Question 1

Given $$f(x) = x - 1$$, $$g(x) = \frac{1}{x + 1}$$, and $$h(x) = x^2$ + 2$$, what is $$(f \circ g \circ h)(1)$$?

Accepted Answer

$$f(g(h(1))) = f(g(3)) = f\left(\frac{1}{4}\right) = \frac{1}{4} - 1 = -\frac{3}{4}$$

Question 2

Find the natural domain of the function $$f(x) = \frac{1}{\sqrt{x} + 1}.$$

Accepted Answer

$$x \geq 0$$

Question 3

Given $$f(x) = 1 + \sqrt{x}$$ and $$g(x) = 3 - x$$, find $$(f \circ g)(x).$$

Accepted Answer

$$(f \circ g)(x) = 1 + \sqrt{3 - x}$$

Question 4

What is the range of the function $$f(x) = x^2$ + \sqrt{x} - 2$$ for $$x \geq 0$$?

Accepted Answer

$$[-2, \infty)$$

Question 5

Express the area $$A of a $$circle in terms of its circumference $$x.$$

Accepted Answer

$$A = \frac{x^2$}{4\pi}$$

Question 6

A cube has surface area $$x. $$What is the length $$d of $$its central diagonal in terms of $$x$$?

Accepted Answer

$$d = \sqrt{3} \left(\frac{x}{6}\$$right)^{1/2}$$

Question 7

Which of the following functions is even?

Accepted Answer

$$h(x) = x^4$ \sqrt{x^2$} + 1$$

Question 8

Given the piecewise function $$f(x) = \begin{cases} x - 1, & x \in [-1, 1] \ 2 - x, & x \in (1, 3] \end{cases}$$, what is $$f(2)$$?

Accepted Answer

$$f(2) = 1$$

Question 9

Which of the following is the correct formula for a vertical shift upward by 3 units of $$y = -x^2$?

Accepted Answer

$$y = -x^2$ + 3$$

Question 10

Suppose $$y = 1 + \frac{1}{5x}. $$What is the domain of this function?

Accepted Answer

$$x 
eq 0$$

Question 11

Given $$f(x) = x^4$ + 2x + 1$$, determine if $$f is $$even, odd, or neither.

Accepted Answer

Neither

Question 12

If $$f(x) = x^2$ - 2x - 3$$, what is the vertex of the parabola?

Accepted Answer

$$(1, -4)$$

Question 13

Which transformation produces $$y = 1 - \sqrt{2x + 4}$$ from $$y = \sqrt{x}$$?

Accepted Answer

Vertical reflection, vertical shift up by 1, horizontal compression by $$\frac{1}{2}$$, and left shift by 2

Question 14

Given $$f(x) = x^3$ - 4x$$, determine if $$f is $$even, odd, or neither.

Accepted Answer

Odd

Question 15

If $$f(x) = x - 1$$ and $$g(x) = \frac{1}{x + 1}$$, what is $$g(f(x))$$?

Accepted Answer

$$g(f(x)) = \frac{1}{x}$$

Question 16

Which of the following best describes the graph of $$y = -\sqrt{-x + 2}$$?

Accepted Answer

A leftward-opening upper half parabola, reflected over the $$x-$$axis and shifted left by 2

Question 17

Given $$f(x) = x^4$ \sqrt{x^2$} + 1$$, is $$f$$ even, odd, or neither?

Accepted Answer

Even

Question 18

If $$f(x) = x - 1$$ and $$g(x) = \frac{1}{x + 1}$$, what is $$f(g(x))$$?

Accepted Answer

$$f(g(x)) = \frac{1}{x + 1} - 1$$

Question 19

Suppose $$f(x) = x^2$ + \sqrt{x} - 2. $$What is the minimum value of $$f(x)$$ for $$x \geq 0$$?

Accepted Answer

$$-2$$

Practice Guidance for Precalculus and Calculus Foundations

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