Functions Toolkit (DP IB Analysis & Approaches (AA): HL): Exam Questions

The functions $f$ and $g$ are defined for $x\in \mathrm{ℝ}$ by $f\left(x\right)=x^2$ and $g\left(x\right)=4x-3$.

Write down the range of $f$.

Find

(i) $\left(f\circ g\right)\left(x\right)$

(ii) $\left(g\circ f\right)\left(x\right)$.

Solve the equation $f\left(x\right)=g\left(x\right)$.

The function $f$ is defined as

$f\left(x\right)=\frac{x^2+1}{x^2},x\in \mathrm{ℝ},x\ne 0$

Show that $f\left(x\right)$ can be written in the form

$f\left(x\right)=1+\frac{1}{x^2}$

Explain why the inverse of $f\left(x\right)$ does not exist and suggest an adaption to its domain so that the inverse does exist.

The domain of $f\left(x\right)$ is changed to $x>0$. Find an expression for $f^{-1}\left(x\right)$ and state its domain and range.

The functions $f$ and $g$ are defined for $x\in \mathrm{ℝ}$ by $f\left(x\right)=\frac{1}{2}\left(4x-3\right)$ and $g\left(x\right)=0.5x+0.75$.

Find

(i) $\left(f\circ g\right)\left(x\right)$

(ii) $\left(g\circ f\right)\left(x\right)$.

Hence write down $f^{-1}\left(x\right)$, and state its domain and range.

The function $f$ is defined by $f\left(x\right)=54x-13$, for $-2<x<20$.

Find $f\left(\frac{5}{2}\right)$.

Find the range of $f$.

Find the value of $f^{-1}\left(122\right)$.

Write down the range of $f^{-1}$.

Consider the function $f\left(x\right)=-6x-3$, for $-5\le x\le 3$.

Find

(i) $f\left(2\right)$

(ii) the value of $x$ such that $f\left(x\right)=15$.

Find the range of $f$.

Write down the domain of $f^{-1}$.

Let $f\left(x\right)=\frac{2x-5}{x+8}$, where $x\ne a$, $x\in \mathrm{ℝ}$.

Write down

(i) the value of $a$

(ii) the range of $f$.

For the graph of $f$, write down the equation of

(i) the vertical asymptote

(ii) the horizontal asymptote.

Find $f^{-1}\left(x\right)$.

For the graph of $f^{-1}$, find the equation of

(i) the horizontal asymptote

(ii) the vertical asymptote.

Prove that the sum of two odd functions is also an odd function.

Let $f\left(x\right)=\frac{{\pi }^2}{x}$, where $x\ne 0$, $x\in \mathrm{ℝ}$.

Show that $f\left(x\right)$ is a self-inverse function.

Let $g\left(x\right)=\frac{-x-2}{5x+1}$, where $x\ne p$, $x\in \mathrm{ℝ}$.

Write down the value of $p$.

Show that $g\left(x\right)$ is a self-inverse function.

Consider the function $f$ defined by $f\left(x\right)=2x^3+3x^2-36x+7$, $x\in \mathrm{ℝ}$.

Sketch the graph of $f$. Clearly label the points where the graph intersects the axes, along with any points that are local maxima or minima.

Let the function $g$ be defined by $g\left(x\right)=2x^3+3x^2-36x+7$, $x\le p$.

Given that $g$ has an inverse:

(i) Write down the largest possible value of $p$

(ii) Find the domain of $g^{-1}$ for the value of $p$ found in part (b)(i)

(iii) Find the value of $g^{-1}\left(0\right)$.

Let the function $h$ be defined by $h\left(x\right)=2x^3+3x^2-36x+7$, $x\ge q$.

Given that $h$ has an inverse:

(i) Write down the smallest possible value of $q$

(ii) Find the domain of $h^{-1}$ for the value of $q$ found in part (c)(i)

(iii) Find the value of $h^{-1}\left(0\right)$.

Let $f\left(x\right)=\frac{5-x^2}{3}$ and $g\left(x\right)=4-\frac{3}{x}$, where each function has its largest possible domain.

Write down the range of $f$.

Write down the domain and range of $g$.

Find

(i) $\left(f\circ g\right)\left(x\right)$

(ii) $\left(g\circ f\right)\left(x\right)$.

Solve the equation $\left(f\circ g\right)\left(x\right)=\left(g\circ f\right)\left(x\right)$.

The functions $f$ and $g$ are defined such that $f\left(x\right)=2x^2-4x$ and $g\left(x\right)=\frac{5x+12}{2}$.

Find $\left(g\circ f\right)\left(x\right)$, giving your answer in the form $\left(g\circ f\right)\left(x\right)=m{\left(x-h\right)}^2+k$, where $m$, $h$ and $k$ are constants to be found.

Hence, or otherwise, find the coordinates of the vertex of the graph of  $y=(g\circ f)\left(x\right)$.

Find $\left(f\circ g\right)\left(x\right)$, giving your answer in the form $\left(f\circ g\right)\left(x\right)=a{x}^2+bx+c$ where $a,b$ and $c$ are constants to be found.

Hence, or otherwise, find the coordinates of the $y$-intercept of the graph of $y=\left(f\circ g\right)\left(x\right)$.

The function $f$ is defined by $f\left(x\right)=\sqrt[3]{4\left(1-x\right)}$, for $-1\le x\le 17$.

Find the range of $f$.

Find $f^{-1}\left(x\right)$.

Write down the domain and range of $f^{-1}$.

The following diagram shows the graph of $y=f\left(x\right)$, for $-3\le x\le 3$. The graph passes through $\mathrm{A}\left(-3,2.5\right)$, $\mathrm{B}\left(3,-2.5\right)$ and $\left(2,0\right)$.

The function $f$ is made up of two linear pieces, the first for $-3\le x\le 2$.

Find $f\left(x\right)$ as a piecewise function.

On the grid above, sketch the graph of $y=f^{-1}\left(x\right)$.

A rectangle has length $4x$ and width $x$.

Find an expression, in terms of $x$, for

(i) the perimeter $P$ of the rectangle

(ii) the area $A$ of the rectangle.

Show that $P\left(A\right)=5\sqrt{A}$.

The graph of $P$, for $0\le A\le 4$, is shown on the following grid.

On the same grid, sketch the graph of $P^{-1}$.

Show that the function $f$, defined by $f\left(x\right)=\text{ln}\left(\frac{{\mathrm{e}}^x+1}{{\mathrm{e}}^x-1}\right)$, is a self-inverse function.

The perimeter, $P$, and the area, $A$, of a square with sides of length $x$ are given by $P=4x$ and $A=x^2$.

Find an expression for

(i) $P$ in terms of $A$

(ii) $A$ in terms of $P$.

Given that $P^{-1}\left(40\right)=A\left(k\right)$, find the value of $k$.

Let $f\left(x\right)=\sqrt{x-14}$, for $x\ge 14$.

Find $f^{-1}\left(2\right)$.

Let $g$ be a function such that $g^{-1}$ exists for all real numbers.

Given that $g\left(14\right)=3$, find $\left(f\circ g^{-1}\right)\left(3\right)$.

Let $f\left(x\right)=\sqrt{2x^2-16x+41}$, where $f$ has its largest possible domain.

Find the domain and range of $f$.

(i) Find the values of $x$ for which $f\left(x\right)=\sqrt{11}$.

(ii) Hence explain why $f$ does not have an inverse function.

Let $f\left(x\right)=x^2-9$ and $g\left(x\right)=x^2-1$, for $x\ge 0$.

Find

(i) $f^{-1}\left(x\right)$

(ii) $g^{-1}\left(x\right)$.

Find $\left(f\circ g\right)\left(x\right)$, giving your answer in the form $a{x}^4+b{x}^2+c$.

Solve the equation $(f\circ g)\left(x\right)=0$.

Express $x^2+12x+24$ in the form ${\left(x+h\right)}^2+k$, where $h,k\in \mathrm{\mathbb{Z}}$.

Let $g\left(x\right)=x+6$.

Given that $\left(f\circ g\right)\left(x\right)=x^2+12x+24$, find $f\left(x\right)$.

Write $2x^2+8x-3$ in the form $a{\left(x+h\right)}^2+k$.

Explain why the function $f$  defined by $f\left(x\right)=2x^2+8x-3,x\in \mathrm{ℝ}$,  does not have an inverse.

The function $g$ defined by   $g\left(x\right)=2x^2+8x-3,x\ge p$  has an inverse.

  (i)      Write down the smallest possible value of $p$.

Given that $p$ takes its smallest possible value:

   (ii)     Find the domain and range of $g^{-1}$.

   (iii)    Find the inverse function $g^{-1}$ .

Solve $(g\circ f)\left(x\right)=21$.

Let $f\left(x\right)$ be an even function and let $g\left(x\right)$ be an odd function. Both functions are defined for all real values of $x$.

Prove the following statements:

(i) $p\left(x\right)=f\left(x\right)g\left(x\right)$ is an odd function.

(ii) $q\left(x\right)=\left(f\circ g\right)\left(x\right)$ is an even function.

Determine whether or not it is possible for the function $r$ defined by

$r\left(x\right)=f\left(x\right)+g\left(x\right)$

to be even or odd, being sure to state clearly any conditions that apply.

The function $f$ is defined by $f\left(x\right)=\frac{ax+b}{cx+d}$, where $a$, $b$, $c$ and $d$ are real constants with $c\ne 0$.

Given that $f$ is a self-inverse function, find the value of $a+d$.

The functions $f$ and $g$ are defined for $x\in \mathrm{ℝ}$ by $f\left(x\right)=3x^2+10x+7$ and $g\left(x\right)=x+d$, where $d\in \mathrm{ℝ}$.

Find the range of $f$.

Given that $\left(g\circ f\right)\left(x\right)>0$ for all $x\in \mathrm{ℝ}$, determine the set of possible values of $d$.

The function $h$ is defined by $h\left(x\right)=-4x^2+24x+8$, for $x\in \mathrm{ℝ}$.

Express $h\left(x\right)$ in the form $a{\left(x-p\right)}^2+q$, where $a,p,q\in \mathrm{\mathbb{Z}}$.

Let $f\left(x\right)={\left(x-3\right)}^2$.

Given that $\left(g\circ f\right)\left(x\right)=h\left(x\right)$, find $g\left(x\right)$.

The functions $f$ and $g$ are defined for $x\in \mathrm{ℝ}$ by $f\left(x\right)=9x-3x^2-3$ and $g\left(x\right)=-\frac{66+2x}{3}$.

Find $\left(g\circ f\right)\left(x\right)$, giving your answer in the form $a\left(x-p\right)\left(x-q\right)$.

Hence write down the $x$-intercepts of the graph of $y=\left(g\circ f\right)\left(x\right)$.

Let $h\left(x\right)=1-2x$, for $x\in \mathrm{ℝ}$.

The graph of $y=\left(f\circ h\right)\left(x\right)$ meets the $y$-axis at $\mathrm{A}$. The graph of $y=\left(g\circ f\right)\left(x\right)$ meets the positive $x$-axis at $\mathrm{B}$.

Find the exact value of $\mathrm{AB}$.

The functions $f$ and $g$ are defined for $x\in \mathrm{ℝ}$ by $f\left(x\right)=\frac{3-2x}{5}$ and $g\left(x\right)=4x-7$.

Find, in the form $mx+c$,

(i) $\left(g\circ f\right)\left(x\right)$

(ii) $\left(f\circ g\right)\left(x\right)$.

Describe a single transformation that maps the graph of $y=\left(g\circ f\right)\left(x\right)$ onto the graph of $y=\left(f\circ g\right)\left(x\right)$.

Given that ${\left(g\circ f\right)}^{-1}\left(p\right)=2$,  find the value of $p$.

Let $f\left(x\right)=\sqrt{-3x^2+8x+16}$, where $f$ has its largest possible domain.

Write down the coordinates of the $y$-intercept of the graph of $y=f\left(x\right)$.

Find the domain and range of $f$.

Let the function $f$ be such that $f\left(x\right)=\sqrt{5x^2-11x+6.05}$.

Find the largest domain of the form $x\ge a$ for which $f^{-1}$ exists, and write down the corresponding range of $f$.

For the domain found in part (a), find $f^{-1}\left(\sqrt{22.05}\right)$.

Let the function $f$ be defined by $f\left(x\right)={\left(2x^2-5x-12\right)}^{-\frac{1}{2}}-k$, where $k$ is a constant and $f$ has the largest possible domain.

Find the domain of $f$.

Given that $\underset{x\to \infty }{\text{lim}}f\left(x\right)=-7$, find the value of $k$.

Write down the equations of any vertical and/or horizontal asymptotes on the graph of $y=f\left(x\right)$.

Consider the function $f$ defined by $f\left(x\right)=x^2-6x+10$, $x\le p$, where $p$ is the largest value such that $f$ has an inverse.

(i) Write down the value of $p$.

(ii) On the same set of axes, sketch the graphs of $f$ and $f^{-1}$.

(iii) Write down the domain and range of $f^{-1}$.

Find the inverse function $f^{-1}$ .

Let the function $g$ be defined by $g\left(x\right)=x^2-6x+10$, $x\in \mathrm{ℝ}$.

(i) Solve $\left(g\circ f\right)\left(x\right)=2$.

(ii) Solve $\left(f\circ g\right)\left(x\right)=2$.

A part of the graph of the function $f\left(x\right)=2x^3-3x^2-12x+8,x\in \mathrm{ℝ}$ is shown below.  

Explain why $f$  does not have an inverse.

The domain of $f$ is now restricted to   $a\le x\le b$  where   $a<0$  and  $b>0$. $a$ and $b$ are chosen so that $f$ has an inverse and the interval $[a,b]$ is as large as possible. 

Find the domain and range of $f^{-1}$