Math Core

This test covers function notation, domain and range, rates of change and symmetry, transformations, combining and composing functions, inverse functions, and piecewise functions.

Question 1

Let $f(x) = x^2 - 3x$ . Find $f(-2)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 2

Find the domain of $f(x) = \sqrt{10 - 2x}$ . Write it as an inequality in $x$ .

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Question 3

Find the average rate of change of $f(x) = x^3$ from $x = 1$ to $x = 3$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 4

Is $f(x) = x^4 - 3x^2$ even, odd or neither?

Question 5

The point $(4, 6)$ is on the graph of $f$ . What point must be on the graph of $g(x) = 2f(x - 3) + 1$ ?

Enter a point like (2, -3)

Question 6

How does the graph of $g(x) = f(3x - 6)$ compare with the graph of $f$ ?

Question 7

A function $f$ has range $[-2, 4]$ . What is the range of $g(x) = -f(x) + 3$ ? Write it as a compound inequality in $y$ .

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Question 8

Let $f(x) = 2x + 5$ and $g(x) = x^2$ . Find a formula for $f\big(g(x)\big)$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 9

Let $f(x) = \sqrt{x + 1}$ and $g(x) = x^2 - 1$ . Find $f\big(g(3)\big)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 10

Let $f(x) = \dfrac{1}{x - 1}$ and $g(x) = x^2$ . What is the domain of $f \circ g$ ?

Question 11

Find $f^{-1}(x)$ for $f(x) = \dfrac{3x - 2}{x + 1}$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 12

Let $f(x) = (x + 1)^2 - 4$ with domain $x \ge -1$ . Find $f^{-1}(5)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 13

Let

f(x) = \begin{cases} 3 - x, & x \le 2 \\ x^2 - 3, & x > 2 \end{cases}

Find $f(2) + f(3)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 14

Find the value of $k$ that makes the graph of this function have no break:

f(x) = \begin{cases} x + k, & x < 3 \\ kx - 1, & x \ge 3 \end{cases}

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 15

Let $f(x) = \lfloor 2x \rfloor$ , where $\lfloor u \rfloor$ is the greatest integer less than or equal to $u$ . Find $f(-1.3)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.