Mathematics Functions and Calculus Flashcards

What is the domain of f(x) = √(4 − x)?

Answer: x ≤ 4

For the square root to be defined, the expression inside must be non-negative: 4 − x ≥ 0 gives x ≤ 4.

What is the derivative of f(x) = 3x² − 2x + 5?

Answer: 6x − 2

Using the power rule: the derivative of 3x² is 6x, the derivative of −2x is −2, and constants vanish, giving 6x − 2.

What is lim(x→2) [(x² − 4) / (x − 2)]?

Answer: 4

Factoring gives (x+2)(x−2)/(x−2) = x+2; substituting x=2 yields 4.

If f(x) = 2x + 1 and g(x) = x², what is f(g(3))?

Answer: 19

g(3) = 9, then f(9) = 2(9) + 1 = 19.

The range of f(x) = −x² + 3 is:

Answer: y ≤ 3

The parabola opens downward with vertex at (0, 3), so the maximum output is 3 and f(x) ≤ 3 for all x.

The gradient of the tangent to f(x) = x³ at x = 2 is:

Answer: 12

f′(x) = 3x², so f′(2) = 3(4) = 12.

Which set of ordered pairs represents a function?

Answer: {(1,2),(2,3),(3,4)}

A function requires each x-value to map to exactly one y-value; only {(1,2),(2,3),(3,4)} satisfies this.