Question 1

Which of the following relations represents a function from the set of real numbers to real numbers?

Accepted Answer

The relation $$y = 2x + 3$$

Question 2

Given the function $$f(x) = \frac{1}{x-2}$$, what is the domain of $$f$$?

Accepted Answer

All real numbers except $$x = 2$$

Question 3

Evaluate the piecewise function $$f(x) = \begin{cases} x^2$ & 	ext{if } x \u003c 0 \ 2x + 1 & 	ext{if } x \geq 0 \end{cases} at$$ $$x = -3.$$

Accepted Answer

$$9$$

Question 4

Which of the following graphs passes the vertical line test and therefore represents $$y as a $$function of $$x$$?

Accepted Answer

A straight line with positive slope

Question 5

Given $$f(x) = x^2$ - 3x + 2$$, find the difference quotient $$\frac{f(x+h) - f(x)}{h}.$$

Accepted Answer

$$2x - 3 + h$$

Question 6

Which transformation will shift the graph of $$f(x) = x^2$ two units up and three units to the right?

Accepted Answer

$$f(x) = (x-3)^2$ + 2$$

Question 7

Let $$f(x) = \sqrt{x}$$ and $$g(x) = x-4. $$What is the domain of $$f(g(x))$$?

Accepted Answer

All real numbers $$x$$ such that $$x \geq 4$$

Question 8

Given $$f(x) = x^3$ and g(x$$) = $$-x^3$$, determine whether $$f is $$even, odd, or neither.

Accepted Answer

Odd

Question 9

Find the sum of the complex numbers $$z_1 = 3 + 2i$ and z_2$ = -1 + 5i.$$

Accepted Answer

$$2 + 7i$$

Question 10

Solve the quadratic equation $$x^2 - 5x + 6 = 0$ using factoring.

Accepted Answer

$$x = 2$$ and $$x = 3$$

Question 11

A rectangular garden is to be fenced on three sides (one side is against a wall). If you have 60 meters of fencing, what dimensions maximize the area?

Accepted Answer

Width $$= 20 m$$, Length $$= 20 m$$

Question 12

What is the end behavior of the polynomial $$f(x) = -2x^4$ + 5x^2$ - 3 as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o -\infty$$

Question 13

Given the polynomial $$p(x) = x^3$ - 4x^2$ + 4x$$, find all its zeros.

Accepted Answer

$$x = 0, x = 2, x = 2$$

Question 14

Use the Linear Factorization Theorem to write a possible formula for a polynomial with zeros at $$x = 1$$, $$x = -2$$, and $$x = 3.$$

Accepted Answer

$$f(x) = (x-1)(x+2)(x-3)$$

Question 15

Divide $$f(x) = x^3$ + 2x^2$ - x + 4 by$$ $$x + 1$$ using synthetic division. What is the remainder?

Accepted Answer

$$2$$

Question 16

Use the Rational Root Theorem to list all possible rational roots of $$f(x) = 2x^3$ - 3x^2$ + x - 6.$$

Accepted Answer

$$\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}$$

Question 17

Given $$f(x) = 2x + 5$$, find the inverse function $$f^{-1}(x$$)$$.

Accepted Answer

$$f^{-1}(x$$) = \frac{x-5}{2}$$

Question 18

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, what is the domain of g(f(x$$))$$?

Accepted Answer

All real numbers

Question 19

Which of the following equations defines $$y as a $$function of $$x$$?

Accepted Answer

$$y = \sqrt{x}$$

Precalculus Exam 1 Study Guide: Functions, Quadratics, and Polynomials

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