Question 1

Which of the following best describes the domain of the function $$f(x) = \frac{1}{x^2$ - 4}$$?

Accepted Answer

All real numbers except $$x = 2$$ and $$x = -2$$

Question 2

Given the quadratic function $$f(x) = 2x^2$ - 8x + 3$$, what is the $$x-$$coordinate of its vertex?

Accepted Answer

$$x = 2$$

Question 3

If $$f(x) = x^2$ - 6x + 8$$, what are the $$x-$$intercepts of the parabola?

Accepted Answer

$$x = 2$$ and $$x = 4$$

Question 4

Which of the following statements about rational functions is TRUE?

Accepted Answer

A rational function can have holes in its graph where both the numerator and denominator are zero at the same $$x$$ value.

Question 5

What is the horizontal asymptote of the function $$f(x) = \frac{3x^2$ + 2x}{x^2$ - 5}$$?

Accepted Answer

$$y = 3$$

Question 6

A rational function $$f(x) = \frac{x^2$ - 9}{x^2$ - 6x + 9} is $$simplified. What is the domain of the simplified function?

Accepted Answer

All real numbers except $$x = 3$$

Question 7

Which of the following is a correct application of Descartes' Rule of Signs?

Accepted Answer

It predicts the number of positive real roots of a polynomial by counting sign changes in the coefficients.

Question 8

If $$f(x) = \frac{5 + x^2$ + \frac{1}{x} + \frac{1}{x-3}}$$, which values of $$x$$ are NOT in the domain?

Accepted Answer

$$x = 0$$ and $$x = 3$$

Question 9

Which statement about the roots of polynomials is correct?

Accepted Answer

If $$a i is a $$root, then $$-a i is $$also a root.

Question 10

Given the polynomial $$f(x) = x^3$ - 4x^2$ + x + 6$$, which of the following is a possible rational root according to the Rational Root Theorem?

Accepted Answer

$$x = 1$$

Question 11

A parabola opens upward and has its vertex at $$(2, -3). $$Which of the following could be its equation?

Accepted Answer

$$y = (x - 2)^2$ - 3$$

Question 12

Which of the following is the result of expanding $$(x - 5)(x + 2)$$?

Accepted Answer

$$x^2 - 3x - 10$

Question 13

If a rational function has a hole at $$x = 2$$, what does this mean?

Accepted Answer

Both the numerator and denominator are zero at $$x = 2$$, and the factor cancels.

Question 14

Which of the following is the correct definition of the range of a function?

Accepted Answer

The set of all possible output values ($$y$$ values) of the function.

Question 15

Synthetic division is most useful for which of the following tasks?

Accepted Answer

Dividing a polynomial by a linear factor of the form $$x - c$$

Question 16

If $$f(x) = x^2$ + 4x + 5$$, what is the minimum value of $$f(x)$$?

Accepted Answer

$$1$$

Question 17

A rational function $$f(x) = \frac{x^2$ - 4}{x^2$ - x - 6} is $$simplified. What are the vertical asymptotes?

Accepted Answer

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

Question 18

Which of the following is a real-world application of rational functions?

Accepted Answer

Modeling the concentration of a drug in the bloodstream over time

Question 19

If $$f(x) = x^2$ - 4x + 4$$, what is the $$y-$$intercept of the parabola?

Accepted Answer

$$y = 4$$

Question 20

Which theorem helps determine the possible rational roots of a polynomial with integer coefficients?

Accepted Answer

Rational Root Theorem

Precalculus Study Notes: Functions, Parabolas, and Rational Functions

Study Guide - Practice Questions