$f^{-1}(x) = \frac{5x+3}{2x-1}$

$[0, \infty)$ for $x \leq -2$ or $x \geq 2$

$(-\infty, 4]$ for $x \leq 3$, but the minimum is at $x = 1.5$

Question 1

Which of the following is the correct domain for the function $f(x) = \sqrt{x - 3} + 1$?

Accepted Answer

$x \geq 3$

Question 2

Given $f(x) = 3x - 2$, what is the inverse function $f^{-1}(x)$?

Accepted Answer

$f^{-1}(x) = \frac{x + 2}{3}$

Question 3

Which of the following statements about the range of $f(x) = x^2 - 3x + 7$ for $x \in [-2, 4]$ is correct?

Accepted Answer

The range is $[-7, 5]$ (excluding $-7$ and including $5$)

Question 4

If $f(x) = |x^2 - 3x - 4|$, which of the following describes how to sketch its graph?

Accepted Answer

Sketch $y = x^2 - 3x - 4$ and reflect the portion below the $x$-axis above the $x$-axis.

Question 5

Which of the following is NOT a function?

Accepted Answer

$f: x \mapsto \pm \sqrt{x}$, \quad x \geq 0$

Question 6

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

Accepted Answer

$(x + 1)^2$

Question 7

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

Accepted Answer

$2x^2 + 1$

Question 8

Which of the following best describes the graph of $y = |x^2 - 3x|$?

Accepted Answer

It is the graph of $y = x^2 - 3x$ reflected in the $x$-axis for $y < 0$.

Question 9

Given $f(x) = \frac{x+3}{2x-5}$, what is the inverse function $f^{-1}(x)$?

Accepted Answer

$f^{-1}(x) = \frac{5x+3}{2x-1}$

Question 10

A function $f(x)$ is defined as $f(x) = \sqrt{x-2}$ for $x \geq 2$. What is the range of $f(x)$?

Accepted Answer

$[0, \infty)$

Question 11

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

Accepted Answer

$2x^2 + 1$

Question 12

Which of the following is the correct range for the function $f(x) = \frac{1}{x-1}$?

Accepted Answer

$y \in \mathbb{R}, y 
eq 0$

Question 13

Given $f(x) = x^2 - 4$ and $g(x) = x + 2$, what is $f(g(x))$?

Accepted Answer

$(x + 2)^2 - 4$

Question 14

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

Accepted Answer

$x 
eq 2, x 
eq -2$

Question 15

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

Accepted Answer

$4x^2 + 4x + 1$

Question 16

Which of the following is the correct range for the function $f(x) = \sqrt{x^2 - 4}$?

Accepted Answer

$[0, \infty)$ for $x \leq -2$ or $x \geq 2$

Question 17

If $f(x) = x^2 - 3x + 4$ for $x \leq 3$, what is the range of $f(x)$?

Accepted Answer

$(-\infty, 4]$ for $x \leq 3$, but the minimum is at $x = 1.5$

Question 18

Which of the following is the correct inverse of $f(x) = \frac{2x-1}{x+3}$?

Accepted Answer

$f^{-1}(x) = \frac{3x+1}{2-x}$

Question 19

A function $f(x)$ is defined as $f(x) = \sqrt{9 - x^2}$ for $x \in [-3, 3]$. What is the range of $f(x)$?

Accepted Answer

$[0, 3]$

Pure Mathematics 3 (Edexcel IAL) - Calculus and Functions Study Notes

Study Guide - Practice Questions