Question 1

Which of the following best describes a function?

Accepted Answer

A relation that assigns each input to exactly one output.

Question 2

Given the function $$f(x) = x^2$ + 1$$, what is its domain $$D$$ and range $$R$$?

Accepted Answer

$$D = \mathbb{R},\ R = [1, \infty)$$

Question 3

Which of the following graphs passes the vertical line test and could represent a function?

Accepted Answer

A graph where every vertical line intersects the curve at most once.

Question 4

Find the domain of the function $$f(x) = \sqrt{x - 5}.$$

Accepted Answer

$$[5, \infty)$$

Question 5

Suppose $$f(x) = 4$$ for all $$x$$ and $$g(x) = x^2$ - 3. $$What is $$(g \circ f)(2)$$?

Accepted Answer

13

Question 6

Which of the following is a property of an even function $$f(x)$$?

Accepted Answer

$$f(-x) = f(x)$$ for all $$x in $$the domain

Question 7

Which transformation is described by $$y = f(x) + c$$?

Accepted Answer

Shift the graph of $$f(x) up by$$ $$c$$ units.

Question 8

Given $$y = -2f(3x) + 1$$, which of the following describes the sequence of transformations applied to $$f(x)$$?

Accepted Answer

Horizontal compression by $$\frac{1}{3}$$, vertical reflection, vertical stretch by $$2$$, and shift up by $$1.$$

Question 9

Which of the following statements about inverse functions is TRUE?

Accepted Answer

A function has an inverse if and only if it is one-to-one (passes the horizontal line test).

Question 10

If $$f(x) is $$invertible, which of the following is always true for its inverse $$f^{-1}(x$$)$$?

Accepted Answer

$$f(f^{-1}(x)) = x$ for all x$ in the domain of f$$^{-1}$$

Question 11

Suppose the concentration of air pollution $$P (in $$ppm) at a distance $$x (in $$meters) from a highway is given by a decreasing function $$P(x). If a $$daycare must be located where $$P(x) \leq 50$$, what does the solution to $$P(x) = 50$$ represent?

Accepted Answer

The minimum distance from the highway where the pollution is at most 50 ppm.

Question 12

Given $$f(x) = 2x + 1$$, find the inverse function $$f^{-1}(x$$)$$.

Accepted Answer

$$f^{-1}(x$$) = \frac{x - 1}{2}$$

Question 13

Which of the following is a correct way to algebraically find the inverse of a function $$y = f(x)$$?

Accepted Answer

Interchange $$x$$ and $$y$$, then solve for $$y.$$

Question 14

If $$f(x) is an $$odd function, which of the following must be true?

Accepted Answer

$$f(-x) = -f(x)$$ for all $$x in $$the domain

Question 15

A function $$f is $$defined by $$f(x) = |x|. $$Which of the following is true?

Accepted Answer

$$f(x) is $$even

Question 16

Suppose $$f(x) is a $$function whose graph is reflected across the line $$y = x to $$obtain the graph of $$f^{-1}(x$$)$$. What must be true about f(x$$)$$?

Accepted Answer

$$f(x)$$ must be one-to-one (invertible)

Question 17

If $$f(x) = x^2$ and g(x) = x + 1$, what is (f$$ \circ g)(2)$$?

Accepted Answer

9

Question 18

Which of the following is a piecewise-defined function?

Accepted Answer

$$f(x) = \begin{cases} x^2$ & x \u003c 0 \ 2x + 1 & x \geq 0 \end{cases}$$

Question 19

A fabric factory uses the function $$f(L) = L^2$ + k to $$calculate the amount of dye needed for $$L$$ meters of fabric. If $$k$$ represents a fixed cost, what does $$L^2$$ represent in this context?

Accepted Answer

The variable amount of dye needed based on the length of fabric produced.

MAT135 Differential Calculus: Introduction and Functions Study Notes

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