Algebra Function Basic Flashcards
(251 cards)
1
Q
In algebra, how is a function defined in simplest terms?
A
A rule that assigns each input exactly one output.
2
Q
What is the ‘domain’ of a function f?
A
All possible input values (x-values) for which f(x) is defined.
3
Q
What is the ‘codomain’ of a function?
A
The set in which all outputs of the function are allowed to lie.
4
Q
What is the ‘range’ (or image) of a function?
A
All actual output values the function produces from its domain.
5
Q
What does it mean for a relation to ‘fail the vertical line test’?
A
At least one vertical line intersects the graph in more than one point, so it is not a function.
6
Q
Define a ‘one-to-one’ (injective) function.
A
A function where each output is produced by at most one input (distinct inputs → distinct outputs).
7
Q
Define an ‘onto’ (surjective) function.
A
A function whose range equals its entire codomain (every element in the codomain is mapped from some input).
8
Q
What does ‘bijective’ mean for a function?
A
It is both injective and surjective, so it has a perfect one-to-one correspondence between domain and codomain.
9
Q
In function notation, f: A → B, what do A and B represent?
A
A is the domain, B is the codomain.
10
Q
What is the difference between ‘range’ and ‘codomain’?
A
Range is the actual set of outputs. Codomain is the set from which outputs can potentially come.
11
Q
How do you typically find the domain of a function given by an expression?
A
Identify values of x that make the expression undefined or invalid (e.g., dividing by zero, negative radicands for even roots), then exclude them.
12
Q
How do you find the domain of a rational function f(x) = P(x)/Q(x)?
A
Exclude values for which Q(x) = 0 from the real domain.
13
Q
What is the domain of f(x) = √(x - 3) in real numbers?
A
All x ≥ 3.
14
Q
What is the domain of f(x) = 1/(x + 2) in real numbers?
A
All real x except x ≠ -2.
15
Q
If f(x) = √(x + 4) + (1/(x - 2)), how do you determine domain?
A
First x + 4 ≥ 0 → x ≥ -4, and x - 2 ≠ 0 → x ≠ 2. Combine to get x ≥ -4 but x ≠ 2.
16
Q
Give an example of a function with domain all real numbers ℝ.
A
Any polynomial, e.g., f(x) = x² + 1.
17
Q
How do you typically find the range of a function algebraically?
A
Solve y = f(x) for x in terms of y, then determine permissible y-values (sometimes more advanced).
18
Q
What is the range of the function f(x) = x², x ∈ ℝ?
A
All real y ≥ 0.
19
Q
What is the range of f(x) = 2x + 3, x ∈ ℝ?
A
All real numbers (−∞, ∞).
20
Q
For f(x) = 1/x, x ≠ 0, what is its range in real numbers?
A
All real y except y ≠ 0.
21
Q
Define a linear function in one variable.
A
A function of the form f(x) = mx + b, where m and b are constants.
22
Q
Define a constant function.
A
A function where f(x) = c for all x in the domain, c is a constant.
23
Q
What is a polynomial function?
A
A function f(x) = aₙxⁿ + … + a₁x + a₀ where coefficients aᵢ are real numbers and exponents are nonnegative integers.
24
Q
Define a quadratic function.
A
A polynomial function of degree 2: f(x) = ax² + bx + c, with a ≠ 0.
25
What is a cubic function?
A polynomial function of degree 3, e.g. f(x) = ax³ + bx² + cx + d.
26
What is a rational function?
A function that can be written as the ratio of two polynomials, f(x) = P(x)/Q(x).
27
Define a piecewise function.
A function defined by different expressions on different intervals of its domain.
28
Define an exponential function (with base b>0, b≠1).
f(x) = b^x, domain = ℝ, range = (0, ∞).
29
Define a logarithmic function (base b>0, b≠1).
f(x) = log_b(x), domain = (0, ∞), range = ℝ.
30
What is an absolute value function in standard form?
f(x) = |x|, which outputs the nonnegative magnitude of x.
31
What is the signum (sign) function?
sgn(x) = 1 if x>0, 0 if x=0, and -1 if x<0.
32
What is the greatest integer (floor) function?
⌊x⌋ gives the largest integer less than or equal to x.
33
What is the fractional part function?
{x} = x - ⌊x⌋, the ‘decimal part’ of x.
34
Define the step function f(x)=c for intervals, e.g., the Heaviside step function.
Heaviside: H(x)=0 if x<0, 1 if x≥0. It's a piecewise constant function.
35
What is the identity function on ℝ?
f(x) = x for all x in ℝ.
36
What is the zero function on ℝ?
f(x) = 0 for all x in ℝ.
37
Define an ‘even’ function in terms of symmetry.
f(−x) = f(x) for all x in domain; symmetric about the y-axis.
38
Define an ‘odd’ function in terms of symmetry.
f(−x) = −f(x) for all x in domain; symmetric about the origin.
39
Give an example of an even function.
f(x) = x².
40
Give an example of an odd function.
f(x) = x³.
41
What is the vertical line test for a function’s graph?
Any vertical line should intersect the graph at most once for it to be a function.
42
Define a horizontal shift for a function f(x).
g(x) = f(x - h) shifts f(x) h units to the right if h>0, left if h<0.
43
Define a vertical shift for a function f(x).
g(x) = f(x) + k shifts the graph k units up if k>0, down if k<0.
44
Define a horizontal scaling for a function f(x).
g(x) = f(bx). If |b|>1, it compresses horizontally; if 0<|b|<1, it stretches.
45
Define a vertical scaling for a function f(x).
g(x) = a·f(x). If |a|>1, vertical stretch; if 0<|a|<1, vertical shrink.
46
How do you reflect a function across the x-axis?
Use g(x) = −f(x).
47
How do you reflect a function across the y-axis?
Use g(x) = f(−x).
48
If y=f(x) is known, what function form flips the graph over the line y=x (i.e., reflection across y=x)?
The inverse function, x = f(y) solved for y, or y = f⁻¹(x).
49
How is the graph of y=f(x)+3 derived from y=f(x)?
Shift it up by 3 units.
50
How is the graph of y=f(x−2) derived from y=f(x)?
Shift it right by 2 units.
51
Define the sum of two functions f and g.
(f + g)(x) = f(x) + g(x).
52
Define the difference of two functions f and g.
(f − g)(x) = f(x) − g(x).
53
Define the product of two functions f and g.
(fg)(x) = f(x)·g(x).
54
Define the quotient of two functions f and g, g≠0.
(f/g)(x) = f(x)/g(x), domain excludes where g(x)=0.
55
Define the composition (f ∘ g)(x).
f(g(x)). First apply g, then apply f to the result.
56
What is an example of function composition usage?
If f(x)=2x+1 and g(x)=x², then (f ∘ g)(x)=f(x²)=2x²+1.
57
How do you find the domain of (f ∘ g)(x)?
Include only x values in the domain of g for which g(x) is in the domain of f.
58
What property must hold for function composition to make sense?
The range of g must overlap with the domain of f.
59
Is function composition generally commutative?
No; (f ∘ g)(x) usually differs from (g ∘ f)(x).
60
If h(x) = (f + g)(x), how is this typically simplified?
h(x)=f(x)+g(x). Keep domain constraints in mind from both f and g.
61
Define an inverse function f⁻¹ if f is one-to-one.
A function that undoes f, satisfying f⁻¹(f(x))=x and f(f⁻¹(x))=x.
62
What is the key property that allows a function to have an inverse?
The function must be injective (one-to-one).
63
What is the graphical relationship of a function and its inverse?
They are reflections of each other across the line y=x.
64
How do you find an inverse function algebraically?
1) Write y=f(x). 2) Solve for x in terms of y. 3) Switch x and y, get y=f⁻¹(x).
65
What is the domain of f⁻¹ compared to f?
It is the range of f.
66
What is the range of f⁻¹ compared to f?
It is the domain of f.
67
If f(x)=2x+5, find f⁻¹(x).
f⁻¹(x)=(x−5)/2.
68
For f(x)=x³, find f⁻¹(x).
f⁻¹(x)=³√x.
69
If f(x)=|x|, is it invertible over ℝ?
No (it fails injectivity). Usually restricted to x≥0 or x≤0 for an inverse.
70
Is the function f(x)=x² invertible on all real ℝ?
No (it’s not one-to-one). Restrict domain to x≥0 or x≤0 to define an inverse.
71
Define ‘increasing function’ on an interval.
f(x₁) < f(x₂) for all x₁ < x₂ in that interval.
72
Define ‘decreasing function’ on an interval.
f(x₁) > f(x₂) for all x₁ < x₂ in that interval.
73
Define a constant function on an interval.
f(x₁) = f(x₂) for all x₁, x₂ in that interval.
74
What is a local maximum of a function?
A point where f(x₀) ≥ f(x) for x near x₀.
75
What is a local minimum of a function?
A point where f(x₀) ≤ f(x) for x near x₀.
76
Define a continuous function at x=a.
f is continuous at a if lim(x→a) f(x)=f(a).
77
Define a discontinuous function at x=a.
A function that breaks the continuity condition (limit doesn’t exist, or doesn’t equal f(a)).
78
Explain boundedness of a function on an interval.
It’s bounded if there’s a real M with |f(x)| ≤ M for all x in that interval.
79
Define the zero (or root) of a function.
An x-value such that f(x)=0.
80
What is the difference between local and global extrema of a function?
Local is highest/lowest in a neighborhood; global is highest/lowest over the entire domain.
81
What is a piecewise-defined function? Give a simple example.
Defined by multiple rules on different intervals. E.g. f(x)=x if x≥0, −x if x<0.
82
When combining piecewise functions, how do you handle domain overlap?
Match intervals carefully and define sums/products piecewise as well.
83
Explain function iteration, e.g., f²(x).
f²(x) means f(f(x)), applying f to x twice.
84
What is the functional equation approach in basic form?
It’s an equation like f(x+y)=f(x)+f(y), used to deduce function forms (like linear for additive).
85
Define the identity function i(x) on domain D.
i(x)=x for all x ∈ D.
86
Describe how to reflect a function across the line y=k.
Subtract from k: new function is 2k−f(x).
87
Describe how to reflect a function across the line x=h.
Replace x with 2h−x in the function rule.
88
If f is one-to-one, how many times will a horizontal line intersect its graph?
At most once (the horizontal line test).
89
If f is onto a set B, what does that mean about the range of f?
Range = B exactly.
90
Why is f(x)=x² not injective on ℝ?
f(−1)=f(1)=1, so multiple x’s produce the same output.
91
Replace x with 2h−x in the function rule.
92
For a cost function C(q)=fixed cost + variable cost·q, what is the domain?
q≥0 (assuming no negative quantity).
93
If f(t)=distance traveled at time t, what is typically the domain?
t≥0, as negative time is not used (depending on context).
94
In a supply/demand function, p=f(q), domain is q≥0. Why?
Quantity q can’t be negative.
95
If f(x)=celcius from Fahrenheit, domain is all reals. The formula is f(F)=(5/9)(F−32). Range?
All real Celcius values: (−∞, ∞).
96
A parent function might be g(x)=x². Then h(x)=x²+3 is a vertical shift. Which direction by how much?
Up 3 units.
97
An inverse variation model is f(x)=k/x. If x doubles, how does f change?
f is halved, since f(x)∝1/x.
98
If f is even, then the graph is symmetrical about which axis?
The y-axis.
99
If f is odd, then f(0)=0 is typical. Why?
Because f(−0)=−f(0) implies f(0)=−f(0) → f(0)=0.
100
If a function passes the horizontal line test, what property does it have?
It is one-to-one (injective).
101
Give a direct variation function formula that passes through (0,0) with slope 5.
f(x)=5x.
102
What is a function transformation in algebra?
An operation (shifting, scaling, reflecting) that modifies a function’s graph into a new position or shape.
103
Write the general form for a transformed function from a parent f(x).
y = a·f(b(x−h)) + k (combining shift, scale, reflection).
104
In y = f(x) + k, how does k affect the graph?
It shifts the graph vertically: up by k if k>0, down if k<0.
105
In y = f(x−h), how does h affect the graph?
It shifts the graph horizontally: right by h if h>0, left if h<0.
106
For a horizontal shift, is it always x−h in the argument if we move right by h?
Yes. Right shift: x−h with h>0. Left shift: x−(−h)=x+h.
107
What does y = −f(x) do to the parent function’s graph?
Reflects it across the x-axis.
108
What does y = f(−x) do to the parent function’s graph?
Reflects it across the y-axis.
109
Define a vertical stretch by factor a>1.
y = a·f(x), multiplies all y-values by a (makes the graph ‘taller’).
110
Define a vertical compression by factor 0 < a < 1.
y = a·f(x), reduces all y-values, making the graph ‘flatter’.
111
How is a horizontal compression by factor b>1 expressed?
y = f(bx), compresses along x-axis by 1/b.
112
Write the formula for shifting a function f(x) up 4 units.
y = f(x) + 4.
113
Write the formula for shifting a function g(x) down 2 units.
y = g(x) − 2.
114
Write the formula for shifting h(x) right 7 units.
y = h(x−7).
115
Write the formula for shifting p(x) left 5 units.
y = p(x + 5).
116
Explain how the domain changes when shifting horizontally by h.
Domain’s x-values shift by −h, but the shape remains the same. If original domain is D, new domain is D shifted by h.
117
If you have y=f(x)+k, does the domain change from the original function f(x)?
No. Vertical shifts do not affect x-values or domain.
118
In y = (x−3)² + 4, what transformations from y=x² are present?
Right shift by 3, and up shift by 4.
119
If a function’s domain was x≥0, then we apply x→x−2 shift, what is the new domain?
x−2≥0 → x≥2.
120
If y=f(x) is known, how do we shift it left by 2 and down by 3 simultaneously?
y = f(x+2) − 3.
121
Give a single transformation formula for shifting a function f(x) up k units and right h units.
y = f(x−h) + k.
122
What is the formula for reflection across the x-axis of y=f(x)?
y = −f(x).
123
What is the formula for reflection across the y-axis of y=f(x)?
y = f(−x).
124
Which reflection changes sign of y-values: x-axis or y-axis reflection?
x-axis reflection changes y-values’ signs.
125
Which reflection changes sign of x-values in the function argument?
y-axis reflection uses f(−x).
126
How do you reflect across y=k (a horizontal line)?
First shift the function so line y=k becomes x-axis, reflect, then shift back: y=k−(f(x)−k).
127
How do you reflect across x=h (a vertical line)?
Replace x with 2h−x. So y = f(2h−x).
128
Is reflection across y=x the same as inverse function?
Yes, graphically it’s reflection over y=x, but it only forms an inverse if f is one-to-one.
129
How do you reflect f(x)=√x across y-axis?
Use y=f(−x)=√(−x). But domain changes to x≤0.
130
Give the transformation form for reflecting across x-axis and shifting up k.
y = −f(x) + k.
131
If y=f(x) has domain x≥−1, after reflection across x-axis, does domain change?
No, reflection across x-axis doesn’t affect x, so domain remains x≥−1.
132
Write a function that vertically stretches f(x) by factor 3.
g(x) = 3f(x).
133
Write a function that vertically compresses f(x) by factor ½.
g(x) = ½f(x).
134
If y=f(x) is multiplied by a factor a<0, is that also a reflection?
Yes. a negative factor includes reflection across x-axis plus vertical stretch/compression.
135
Compare the transformation y=2f(x) vs. y=f(2x) in plain English.
y=2f(x) is vertical stretch by factor 2; y=f(2x) is horizontal compression by factor 1/2.
136
Which formula changes outputs to half their size: y=0.5f(x) or y=f(0.5x)?
y=0.5f(x) compresses vertically by ½.
137
If f(x)≥0 for x≥0, then 3f(x)≥?
≥0 as well, scaled up 3 times but no sign change.
138
In a vertical stretch by factor a>1, does domain or range expand?
Range is stretched, domain is unaffected.
139
What if y=−2f(x)? Summarize the transformations.
Reflection across x-axis, then vertical stretch by factor 2.
140
If parent function is y=|x|, what is y=3|x| doing?
Vertical stretch by factor 3.
141
If y=f(x) passes (2,4), where does y=4f(x) pass for x=2?
At (2,16).
142
Write a function for horizontally stretching f(x) by factor 2.
g(x)=f(x/2) (the inside is x/2 for a 2× horizontal stretch).
143
Write a function for horizontally compressing f(x) by factor ½.
g(x)=f(x/(1/2))=f(2x).
144
Which transformation is y=f(3x)?
A horizontal compression by factor 1/3.
145
If f(x)=x², then g(x)=f(2x) means g(x)=(2x)²=4x². Which transformation is that?
Horizontal compression by factor ½, but it looks like a vertical stretch by 4 in the equation.
146
If h>1, what does y=f(hx) do?
Horizontal compression by factor 1/h.
147
If 0
Horizontal stretch by factor 1/h (which is >1).
148
Does y=f(bx) for b>0 change the domain set for real x?
No, but the function's features appear ‘compressed or stretched’ horizontally.
149
If y=f(½x), how is the graph changed from y=f(x) at x=4?
We evaluate f(2), so the features shift to the right (it’s a stretch factor 2).
150
General formula for a horizontal scale factor k>0 is y=f((1/k)x). True or false?
True. If k>1, that’s a stretch; if 0
151
In a horizontal stretch by factor 3, a point at x=2 moves to x=6 or x=2/3?
It moves to x=6 (the function that was at 2 is now at 6).
152
What is the general form combining shift, scale, reflection?
y = a·f(b(x−h)) + k.
153
In y=a·f(b(x−h))+k, define each parameter’s effect: a, b, h, k.
a=vertical scale ± reflection, b=horizontal scale ± reflection, h=horizontal shift, k=vertical shift.
154
Which transformations are done first, horizontal shifts or horizontal scalings?
Inside the parentheses: do scaling b then shift h. But practically we rewrite carefully to see the effect.
155
Which transformations are done after you handle the x-changes (like b(x−h))?
Outside transformations: vertical scaling by a, then vertical shift by k.
156
If we have y=−3f(2(x−1))+5, list transformations in order.
1) Shift right 1, 2) Horizontal compression by factor 1/2, 3) Reflect across x-axis, 4) Vertical stretch by 3, 5) Shift up 5.
157
Sometimes we combine reflection across y-axis if b<0 in f(bx). True?
Yes. If b<0, there's also a reflection across the y-axis.
158
If a<0 in a·f(...), do we also have reflection across x-axis?
Yes, negative a includes reflection across x-axis plus scale factor |a|.
159
Give the domain shift for y=f(b(x−h)) if b>0.
Shift to x≥h if original domain was x≥0, then compress or stretch by factor 1/b along x.
160
For y=a·f(b(x−h))+k, how does the range shift if originally it was [m,∞) for f?
Scaled by factor a, reflected if a<0, then shifted up by k.
161
Explain how to rewrite y=a·f(b(x−h))+k if we want to see horizontal shift more clearly.
Inside: b(x−h) = b·x − b·h, so shift is h = (b·h)/b but we have to be mindful that dividing or factoring out b.
162
Define g(x)=−2f(3x+6). Which transformations are inside the parentheses?
3x+6 → factor out 3: 3(x+2). So horizontal shift left 2, then horizontal compression factor 1/3.
163
Continuing that example, what do −2 and +1 do?
−2 is reflection across x-axis plus vertical stretch by factor 2, +1 is shift up by 1.
164
If f(0)=2, what is g(0) for g(x)=2f(x)? Summarize the result.
g(0)=2×f(0)=2×2=4; vertical scale factor 2 at x=0.
165
In y = f(2x−4), we can factor out 2: y = f[2(x−2)]. Summarize transformations.
Shift right 2, then horizontal compression factor 1/2.
166
If we see y = f(−(x−3)), that is f(−x+3). Summarize transformations.
Shift right 3, reflect across y-axis, depending on sign inside. Actually it's f(3−x), so it's shift left 3? Carefully analyze.
167
Wait, f(−(x−3))=f(−x+3). That means x is replaced by (3−x). So is that a reflection across x=1.5 or simpler?
Yes, rewriting might help. But typically we see it as reflection across y-axis if we fix the expression. We must be precise.
168
Key tip: always isolate (x−h) by factoring out b. Then interpret sign for reflection. Good idea?
Yes. Factoring out the coefficient of x is the best approach.
169
If the inside coefficient of x is negative, we get reflection across the y-axis plus possible shift. T/F?
True.
170
If the outside coefficient is negative, we get reflection across the x-axis. T/F?
True.
171
In practice, is it best to handle horizontal transformations first or vertical transformations first?
Typically handle inside (horizontal) transformations first, then the outside (vertical).
172
When we do a horizontal shift x→(x−h), how does domain shift?
All x-values shift by +h. If domain was D, new domain is {x | x−h ∈ D}.
173
When we do a vertical shift up k, how does range shift?
All y-values shift up by k. If range was R, new range is R+k.
174
When we reflect across x-axis, how does range change if originally it was [m,∞)?
New range becomes (−∞, −m].
175
When we reflect across y-axis, does range necessarily change from the original?
No, reflection across y-axis affects domain (x) not range (y).
176
A horizontal scale factor by 1/2 (i.e. y=f(2x)) might do what to domain if the original domain was x≥0?
Now x≥0 is mapped to x≥0 still, but features occur twice as fast.
177
Vertical stretch by factor a>1 modifies range how?
All outputs are multiplied by a, so if range was [m,n], new range is [am, an] (assuming no reflection).
178
If a<0 in a vertical scale, e.g. −3, then there's also reflection. True?
Yes, so the range flips sign plus scaling by 3.
179
If domain was (−∞, 5] for f(x), and we shift right 2, what's new domain for g(x)=f(x−2)?
(−∞, 7].
180
If the range was all real y≥−1, after up shift by 3, it becomes y≥ what?
y≥2.
181
If the range was all real y≥1, after reflection across x-axis, it becomes y≤ −1. T/F?
True.
182
What is the recommended step to handle y = 2f(−(x+1)) − 3 first?
Rewrite inside: −(x+1)=−x−1, factor out negative, etc. Then interpret reflection, shift, scale step by step.
183
In transformations, is the order of horizontal transformations critical?
Yes, the order in which we apply shift vs. scale matters inside the argument.
184
Which is typically done first in y=f(b(x−h)) if we read left to right in the argument?
First shift x→(x−h), then scale x→b(x−h).
185
Can we do a combined transformation as x→b(x−h) in one step if we’re comfortable with factoring b out?
Yes, but carefully to avoid sign mistakes.
186
If we see y=−f(2(x+1))+4, list transformations in a clear sequence.
1) Shift left 1, 2) Horizontal compression by factor 1/2, 3) Reflect across x-axis, 4) Vertical shift up 4.
187
If the function is y=x² originally, how do we get y=−(x−3)²+2 in transformations?
Shift right 3, reflect across x-axis, shift up 2.
188
If we want to compress horizontally by factor 1/3 and shift up 5 from y=x³, how do we write that?
y = x³ becomes y=f(3x)+5 or y=(3x)³+5 = 27x³+5 if f(x)=x³.
189
Are transformations generally commutative (i.e. does shifting then reflecting yield same as reflecting then shifting)?
Not always, we must follow the correct order or rewrite carefully.
190
If we do y=|2x| vs. y=2|x|, which is a horizontal vs vertical change?
y=|2x| is horizontal compression by 1/2. y=2|x| is vertical stretch by 2.
191
List the transformations in y=2f(3x−6)−4, in correct order.
1) Shift right 2 (since 3(x−2)), 2) Horizontal compression factor 1/3, 3) Vertical stretch factor 2, 4) Shift down 4.
192
When f is piecewise, do transformations apply piecewise or overall?
They apply to each piece, adjusting intervals accordingly.
193
If a function is periodic (like sine/cosine), does horizontal scaling change its period?
Yes. y=f(bx) changes period to original period / b.
194
If f is odd, reflection across y-axis vs x-axis – do they produce the same or different shapes?
For an odd function, reflection across y-axis is the negative. But it’s not necessarily the same shape as reflection across x-axis. We must be cautious.
195
‘To compress horizontally by factor k’ means points get closer by factor k or 1/k?
If the formula is y=f(kx), actual numerical factor is 1/k. So if k>1, that’s a compression.
196
Why do we factor out the b from x−h to interpret horizontal transformations?
Because y=f(b(x−h)) is y=f[b(x−(h))], shifting is h units, not h/b. Factoring clarifies.
197
In a function transformation, can reflection be combined with scaling into a single factor a < 0?
Yes, a negative factor handles reflection + scale.
198
Are vertical transformations done after horizontal ones in standard function transformation approaches?
Yes. Typically we handle inside function transformations (horizontal) first, then outside (vertical).
199
If a function has domain [0,∞), after y=f(x−2), the domain is [2,∞). Why?
Because x≥2 ensures x−2≥0 for the new function.
200
If a function has range (−∞,10], and we do y=f(x)+5, new range is (−∞, 15]? T/F?
True, everything shifts up by 5.
201
Final tip: to identify transformations, always rewrite y in the form y=a·f(b(x−h))+k. T/F?
True. That clarifies each parameter’s effect.
202
Find f(3) if f(x) = 2x + 1.
f(3) = 2(3) + 1 = 7.
203
If g(x) = x², compute g(−2).
g(−2) = (−2)² = 4.
204
Let h(x) = 5 − x. Evaluate h(7).
h(7) = 5 − 7 = −2.
205
Find f(0) if f(x) = 3x² + 4.
f(0) = 3(0)² + 4 = 4.
206
If j(x) = x + 1, what is j(j(2))?
First j(2)=2+1=3; then j(3)=3+1=4.
207
For k(x)=|x|, evaluate k(−5).
|−5|=5.
208
If p(x)=x−3, solve p(x)=0.
x−3=0 → x=3.
209
Given r(x)=4−2x, find r(2) + r(−2).
r(2)=4−2(2)=0; r(−2)=4−2(−2)=4+4=8; sum=0+8=8.
210
If m(x)=2x+5, compute m(4)−m(1).
m(4)=2(4)+5=13; m(1)=2(1)+5=7; difference=13−7=6.
211
For f(x)=x²−1, find f(2)+f(−2).
f(2)=2²−1=4−1=3; f(−2)=4−1=3; sum=3+3=6.
212
If f(x)=x+2, for which x does f(x)=0?
x+2=0 → x=−2.
213
If g(x)=2x−3, solve g(x)=1.
2x−3=1 → 2x=4 → x=2.
214
Evaluate f(−1) if f(x)=3−x².
f(−1)=3−(−1)²=3−1=2.
215
Find h(2) if h(x)=3x+5 for x≥2. No domain issue, so h(2)=?
h(2)=3(2)+5=6+5=11.
216
If j(x)=−x, compute j(5).
j(5)=−5.
217
If k(x)=2, a constant function, find k(10).
Always 2.
218
For r(x)=x³, find r(−3).
r(−3)=−27.
219
Given p(x)=4−x, find p(0).
p(0)=4−0=4.
220
Compute q(1)+q(2) if q(x)=2x.
q(1)=2; q(2)=4; sum=6.
221
If f(x)=5x−1, for which x is f(x)=9?
5x−1=9 → 5x=10 → x=2.
222
If f(x)=x², compute f(3)−f(2).
f(3)=9; f(2)=4; difference=5.
223
Find domain & range for g(x)=|x| if no restrictions otherwise.
Domain: all real x. Range: [0,∞).
224
Evaluate h(1)+h(−1) if h(x)=x²+1.
h(1)=1+1=2; h(−1)=1+1=2; sum=4.
225
For j(x)=4−x², find j(0).
j(0)=4−0=4.
226
If m(x)=3x−4, find m(3) and m(−1).
m(3)=3(3)−4=9−4=5; m(−1)=3(−1)−4=−3−4=−7.
227
Compute p(2) for p(x)=2x+3.
p(2)=2(2)+3=7.
228
If f(x)=x−2, find x if f(x)=5.
x−2=5 → x=7.
229
For r(x)=−2x, evaluate r(−4).
r(−4)=−2(−4)=8.
230
If s(x)=x²+1, find s(1) & s(−1).
s(1)=2; s(−1)=2.
231
If f(x)=2x+5, find f(0)+f(−2).
f(0)=5; f(−2)=2(−2)+5=1; sum=6.
232
Let h(x)=2−x. For x=4, h(4)=?
2−4=−2.
233
If k(x)=3−2x, solve k(x)=0.
3−2x=0 → 2x=3 → x=1.5.
234
If j(x)=x+1, find j(0).
1.
235
Compute f(1)+f(2)+f(3) if f(x)=x.
1+2+3=6.
236
If g(x)=x²−4, find g(2).
2²−4=4−4=0.
237
For p(x)=−x, compute p(5). Then sum p(5)+p(−5).
p(5)=−5; p(−5)=5; sum=0.
238
If q(x)=1−2x, find x if q(x)=−3.
1−2x=−3 → −2x=−4 → x=2.
239
Let f(x)=x². Evaluate f(−1)+f(−2).
f(−1)=1; f(−2)=4; sum=5.
240
For r(x)=2x−1, find r(1), r(2), r(3).
r(1)=1; r(2)=3; r(3)=5.
241
If t(x)=|x−3|, evaluate t(3).
|3−3|=0.
242
If u(x)=x²−1, find u(1).
1²−1=0.
243
If h(x)=|x|−2, compute h(−2).
|−2|−2=2−2=0.
244
Let k(x)=3x. Solve k(x)=9.
3x=9 → x=3.
245
For a constant function c(x)=7, find c(−100).
7.
246
If g(x)=x−4, evaluate g(5)+g(−1).
g(5)=1; g(−1)=−5; sum=−4.
247
If f(x)=2−x², find f(2).
2−(2²)=2−4=−2.
248
For p(x)=4|x|, compute p(−3).
4|−3|=12.
249
If q(x)=x+2, find x for which q(x)=0.
x+2=0 → x=−2.
250
Compute r(2)+r(3) for r(x)=2x−5.
r(2)=−1; r(3)=1; sum=0.
251
If t(x)=−x+10, solve t(x)=4.
−x+10=4 → −x=−6 → x=6.
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