Pre-Calculus Flashcards
(33 cards)
Q: What does it mean for two functions to be inverses?
A: If function f takes input a to output b, then its inverse f⁻¹ takes b back to a. They “reverse” each other.
Q: Do all functions have inverses?
A: No. A function must be one-to-one (each output paired with only one input) to have an inverse.
Q: Why is the function h(x) = {1→2, 2→1, 3→2, 4→5} not invertible?
A: Because h⁻¹(2) would map to both 1 and 3, violating the definition of a function.
What is the limit of f(x) = x + 2 as x approaches 3?
A: 5. As x gets closer to 3 from both sides, the y-value approaches 5.
Q: Is the limit the same as the value of the function at that point?
A: Not always. The limit describes behavior near a point, not necessarily the function’s value at that point.
Q: What is the difference between f and g if both approach 5 at x = 3, but g is undefined at x = 3?
A: g(x) has a limit of 5 at x = 3, but g(3) is undefined. Limits describe behavior near x, not at x.
Q: What does it mean when we say a limit equals 5?
A: It means as x gets closer to a value (like 3), f(x) gets arbitrarily close to 5 — from both sides.
Q: What is the notation for the limit of f(x) as x approaches 3?
A: lim(x→3) f(x)
Q: What does lim(x→a) f(x) mean?
A: It means we are finding the value that f(x) approaches as x gets close to a.
Q: What does the “horizontal line test” have to do with limits?
A: It’s used for invertibility, not limits — but limits require the function to approach the same y-value from both sides.
Q: What is a one-sided limit?
A: It’s the value a function approaches as x approaches a point from one side only (left or right).
Q: What does a removable discontinuity look like on a graph?
A: A hole in the graph — the limit exists, but the function is not defined at that point.
Q: What must be true for a limit to exist?
A: The function must approach the same value from both the left and the right as x approaches the point.
Mathematically:
limₓ→a⁻ f(x) = limₓ→a⁺ f(x)
If these are equal, then limₓ→a f(x) exists.
Q: What is a derivative?
A: The derivative measures how a function changes — it’s the instantaneous rate of change or the slope of the tangent line at a point.
Q: What is the notation for a derivative?
A: Common notations include:
f ′(x)
dy/dx
d/dx [f(x)]
Q: What does it mean if the derivative of a function is zero at a point?
A: The function has a horizontal tangent at that point — it could be a local max, min, or saddle point.
Q: What is the Power Rule for derivatives?
A: If f(x) = xⁿ, then f ′(x) = n·xⁿ⁻¹
Q: What does the derivative tell us about the graph of a function?
If f ′(x) > 0 → function is increasing
If f ′(x) < 0 → function is decreasing
If f ′(x) = 0 → possible max/min (check further)
Q: What is the Power Rule for derivatives?
A: If f(x) = xⁿ, then f ′(x) = n·xⁿ⁻¹
Find the derivative of:
f(x) = x² + 3x + 5
Use the Power Rule: f ′(x) = n·xⁿ⁻¹
d/dx[x²] = 2x
d/dx[3x] = 3
d/dx[5] = 0
Q: Find the derivative of:
f(x) = 4x³ − 2x² + 7x − 10
Use the Power Rule: f ′(x) = n·xⁿ⁻¹
d/dx[4x³] = 12x²
d/dx[−2x²] = −4x
d/dx[7x] = 7
d/dx[−10] = 0
Q: Find the derivative of:
f(x) = 5 / x
Rewrite: f(x) = 5x⁻¹
d/dx[5x⁻¹] = 5 × (−1)x⁻² = −5 / x²
Solve
2/3(3/2(x+8)) - 8
2/3(3/2(x+8)) - 8
x+8 - 8
x
Solve
s(x-7/2) +7
2(x-7/2) + 7
x-7 + 7
x
