Functions and Indices Problems Workshop 4

Question 1

Sketch the graph of y = x³. What type of transformation is this?

Answer 1

Base cubic function, no transformations

Question 2

Sketch the graph of y = -x³. What type of transformation is this?

Answer 2

Reflection across the x-axis

Question 3

Sketch the graph of y = x³ - 3. What type of transformation is this?

Answer 3

Translation downwards by 3 units

Question 4

Sketch the graph of y = (x + 2)³. What type of transformation is this?

Answer 4

Translation to the left by 2 units

Question 5

Sketch the graph of y = (x + 2)³ - 4. What type of transformation is this?

Answer 5

Translation left by 2 units and down by 4 units

Question 6

Sketch the graph of y = x(x - 4)². What is the degree of this polynomial?

Answer 6

Degree 3

Question 7

Sketch the graph of f(x) = (x + 1)(x - 2)(x - 3). What is the degree of this polynomial?

Answer 7

Degree 3

Question 8

Find the inverse of f(x) = 3x + 2. What is the domain of the inverse?

Answer 8

f⁻¹(x) = (x - 2)/3, Domain: R

Question 9

Find the inverse of f(x) = x - 2. What is the range of the inverse?

Answer 9

f⁻¹(x) = x + 2, Range: R

Question 10

Find the inverse of f(x) = 1/x. What is the domain of the inverse?

Answer 10

f⁻¹(x) = 1/x, Domain: {x ∈ R | x ≠ 0}

Question 11

Find the inverse of f(x) = 3√x. What is the range of the inverse?

Answer 11

f⁻¹(x) = x³/27, Range: R

Question 12

Simplify 5x⁴ × 4x⁶ using index laws.

Answer 12

20x¹⁰

Question 13

Simplify 14x⁹ ÷ 7x³ using index laws.

Answer 13

2x⁶

Question 14

Simplify 8t⁰ - 5 using index laws.

Answer 14

3

Question 15

Simplify (5p³)³ using index laws.

Answer 15

125p⁹

Question 16

Simplify 3a³b² × 5a⁸b⁵ × 10b³ using index laws.

Answer 16

150a¹¹b¹⁰

Question 17

Simplify 16p⁴q⁷ ÷ -4q²p⁷ using index laws.

Answer 17

-4q⁵/p³

Question 18

Simplify 3a²b³ ÷ 12ab⁵ × 2 using index laws.

Answer 18

a²/16b⁴

Question 19

Simplify 3(xy²)³ × 4x⁴y² ÷ 8x²y using index laws.

Answer 19

3x⁵y⁷/2

Question 20

Express 2x⁻⁴ in positive indices.

Answer 20

2/x⁴

Question 21

Express 3/8a⁻⁵b²c⁻⁷ in positive indices.

Answer 21

3b²c⁷/8a⁵

Question 22

Express -8(x⁻⁵)⁻³ in positive indices.

Answer 22

-8x¹⁵

Question 23

Express 12x⁻²y³ ÷ 15x⁻³y⁻² in positive indices.

Answer 23

4xy⁵/5

Question 24

Write √x in index form.

Answer 24

x¹/²

Question 25

Write 3√a in index form.

Answer 25

a¹/³

Question 26

Write 5√x² in index form.

Answer 26

x²/5

Question 27

Write 321/5 in terms of roots.

Answer 27

5√32

Question 28

Write 163/2 in terms of roots.

Answer 28

√163

Question 29

Write x³/4 in terms of roots.

Answer 29

4√x³

Question 30

Evaluate 27^(1/3) without a calculator.

Answer 30

3

Question 31

Evaluate 64^(-4/3) without a calculator.

Answer 31

1/256

Question 32

Evaluate 8^(27/2) without a calculator.

Answer 32

4

Question 33

Simplify 2x^(1/2) × 5x³/2.

Answer 33

10x²

Question 34

Simplify 12y⁵/4 ÷ 36y³/4.

Answer 34

y^(1/2)/3

Question 35

Simplify (2y^(1/3))⁶.

Answer 35

64y²

Question 36

Simplify 4x^(1/2)/9x^(1/3) × 3/2.

Answer 36

64x^(1/2)

Question 37

What will be a town’s population in five years if it’s initially 50,000 and increases by 12% per year?

Answer 37

88,117

Question 38

Sketch the function f(x) = (x - 1)(x + 1)(x + 2). What are the x-intercepts?

Answer 38

(-2, 0), (-1, 0), (1, 0)

Question 39

What are the coordinates of the y-intercept for f(x) = (x - 1)(x + 1)(x + 2)?

Answer 39

(0, -2)

Question 40

What is the graph of the function y = 4 - x^2?

Answer 40

A downward-opening parabola with vertex at (0, 4) ## Footnote This function represents a quadratic equation.

Question 41

What is the graph of the function y = (x - 2)^2 - 3?

Answer 41

A upward-opening parabola with vertex at (2, -3) ## Footnote This function represents a quadratic equation.

Question 42

What is the graph of the function y = x^3 + 5?

Answer 42

A cubic function that passes through (0, 5) ## Footnote This function represents a cubic equation.

Question 43

What is the graph of the function y = (x + 3)^3 + 5?

Answer 43

A cubic function shifted left by 3 units and up by 5 units ## Footnote This function represents a cubic equation.

Question 44

What is the graph of the function y = √x + 4?

Answer 44

A square root function shifted up by 4 units ## Footnote This function represents a radical equation.

Question 45

What is the graph of the function y = -|x - 1|?

Answer 45

An inverted V-shape with vertex at (1, 0) ## Footnote This function represents an absolute value equation.

Question 46

What is the inverse of the function f(x) = x^2 + 5 with domain D = {x ∈ R | x ≥ 0}?

Answer 46

f^{-1}(x) = √(x - 5); Domain: Df^{-1} = {x ∈ R | x ≥ 5}; Range: Rf^{-1} = {y ∈ R | y ≥ 0} ## Footnote The function is defined for non-negative x.

Question 47

What is the inverse of the function f(x) = 1/(x - 1)?

Answer 47

f^{-1}(x) = 1/x + 1; Domain: Df^{-1} = {x ∈ R | x ≠ 0}; Range: Rf^{-1} = {y ∈ R | y ≠ 1} ## Footnote This function is a rational function.

Question 48

Evaluate 100^1/3

Answer 48

10 ## Footnote This is the cube root of 100.

Question 49

Evaluate 125^{-2/3}

Answer 49

1/25 ## Footnote This is the reciprocal of the cube root of 125 squared.

Question 50

Evaluate 4^{-5/2}

Answer 50

1/32 ## Footnote This is the reciprocal of the square root of 4 raised to the fifth power.

Question 51

Simplify 6y^3/5 × y^7/5

Answer 51

6y^{10/5} = 6y^2 ## Footnote This involves multiplying the coefficients and adding the exponents.

Question 52

Simplify x^3/7y^{-4/5} ÷ x^{-2/7y^3/5}

Answer 52

x^{3 + 2}y^{(3/5 + 4/5)} = x^{5/7}y^{7/5} ## Footnote This involves applying the properties of exponents.

Question 53

Simplify (p^{3/2})^4/7

Answer 53

p^{6/7} ## Footnote This involves multiplying the exponent by 4/7.

Question 54

Simplify 64x^{2 - 1/6} ÷ 32x^{5/2 - 5/5}

Answer 54

2x^{-1/3} ## Footnote This involves simplifying the coefficients and exponents.

Question 55

If a bacterial colony triples in number every day, how many bacteria will be present after 5 days if the initial number is 1000?

Answer 55

243,000 ## Footnote This is calculated using the formula 1000 * 3^5.