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

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

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

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

Write down the range of $f$

Write down the domain and range of $g$.

Find

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

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

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

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

Write down the range of $f$.

Write down an expression for $f^{-1}$.

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

The perimeter, P, and area, A, of a given square can be expressed by $\mathrm{P}=4x$ and $\mathrm{A}=x^{\mathit{2}}$ respectively, where $x$ is the length of the side of the square.

Write down an expression for:

(i) P in terms of A, $P\left(A\right)$

(ii) A in terms of P, $A\left(P\right)$.

${\mathrm{P}}^{-1}\left(40\right)=\mathrm{A}\left(k\right)$

Find the value of $k$and $\mathrm{A}\left(\mathrm{k}\right)$.

The values of two functions, $f$ and $g$, for certain values of $x$ are given in the following table:

Find the value of $f^{-1}\left(8\right)$.

Find the value of $(f\circ g)(-2)$.

Given that $f\left(x\right)$ is a linear function, find $f\left(x\right)$.

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 $(f\circ g^{-1})\left)\right(3)$.

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

Find the domain and range of $f$.

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

(ii)    Use your answer to part (b)(i) to explain why the inverse function $f^{-1}$ does not exist.

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

Find

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

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

Find $(f\circ g)\left(x\right)$ 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 $a(x+b{)}^2+c$, where $a,b,c\in Z$.

Given that $g\left(x\right)=x+6$ and $(f\circ g)\left(x\right)=x^2+12x+24$, find $f\left(x\right)$.

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

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:

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

$q\left(x\right)=(f\circ g)\left(x\right)$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 such that  $f\left(x\right)=9x-3x^2-3$   and $g\left(x\right)=-\frac{66+2x}{3}$  , both for  $x\in \mathrm{ℝ}$ .

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

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

Let $h\left(x\right)=1-2x$.

Find the distance between the $y$-intercept of the graph of  $y=(f\circ h)\left(x\right)$  and the positive $x$-intercept of the graph of  $y=(g\circ f)\left(x\right).$Your answer should be given as an exact value.

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

Given that the inverse function $f^{-1}$ exists, and that the domain of $f$ is as large as possible,

suggest a domain for $f$and write down the corresponding range.

Based on your answer to part (a), find $f^{-1}\left(\sqrt{22.05}\right)$.

Let $f\left(x\right)=\sqrt{-3x^2+8x+16}$ .

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

Given that $f$ has the largest possible valid domain,

find the domain and range of $f$.

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

Find the domain of $f$.

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

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

The following diagram shows the graph of  $y=f\left(x\right)$,  for a function $f$ that has the domain $-3\le x\le 3$.  Point A has coordinates  $(-3,2.5)$ and point B has coordinates $(3,-2.5)$.  The $x$-intercept of the function is $(2,0)$ as shown.

$f$ can be written as a piecewise function, where each of the two pieces is a linear function and where the domain of the first function is $-3\le x\le 2$.

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

Sketch the graph of  $y=f^{-1}\left(x\right)$ on the same grid above.

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

Rewrite $h\left(x\right)$ in the form $a(x+b{)}^2+c$ ,  where $a,b,c\in \mathrm{\mathbb{Z}}$.

Given that  $f\left(x\right)=(x-3{)}^2$ and that $(g\circ f)\left(x\right)=h\left(x\right)$ ,  find $g\left(x\right)$.

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

Giving your answers in the form $y=mx+c$,  find

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

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

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

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

Let the functions $f$ and $g$ be defined by $f\left(x\right)=\frac{9}{4}{x}^2-1$   and  $g\left(x\right)=x^2-2$,  both for $x\ge 0$  .

Find

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

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

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

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

A rectangle has length $l=4x$ and width $w=x$.

Find an expression for

(i) the perimeter of the rectangle, P, in terms of $x$.

(ii) the area of the rectangle, A, in terms of $x$.

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

The graph of the function P, for $0\le A\le 4$ , is shown below.

On the grid above, draw the graph of the inverse function $P^{-1}$

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

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

Consider the function  $f\left(x\right)=a{x}^4+b{x}^3+c{x}^2+dx+e$ where  $a,b,c,d,e\in \mathrm{ℝ}$ .

Show that:

(i) if $f$ is even then $b=d=0$.

(ii) if $f$ is odd then $a=c=e=0$.

Consider the function $g$  defined by   $g\left(x\right)=(3x+p)(x-2)(qx+1)(2x+3)$  where $p$  and $q$  are real constants.

Find the possible values of $p$  and $q$ in the case where $g$ is an even function.

Use proof by contradiction to show that $g$  can never be an odd function.

Consider the function $f$ defined by $f\left(x\right)=\frac{2x-5}{3x+k},x\in R,x\ne -\frac{k}{3}$.

In the case where $f$  is self-inverse, find the value of $k$.

In the case when the graphs of $f$ and $f^{-1}$ intersect at exactly one point, find the possible values of $k$.

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.