Functions Toolkit (DP IB Applications & Interpretation (AI): 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  $(g\circ f)\left(x\right)=m{\left(x-h\right)}^2+k$  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 $(f\circ g)\left(x\right)$,  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 $y=(f\circ g)\left(x\right).$

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 $P=4x$and $A=x^2$ respectively, where x is the length of the side of the square.

Write down an expression for: 

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

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

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

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

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

Find the value of f ^-18.

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

Given that f (x) is a linear function, find f (x).

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

Find f ^-1(2).

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

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

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

(i) Show that ${\left(x+a\right)}^2+b=x^2+2ax+\left(a^2+b\right)$

(ii) Hence show that  $x^2+12x+24={\left(x+6\right)}^2-12$

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

(i) Sketch the graph of the function $f$ defined by $f\left(x\right)=2x^2+8x-3,x\in \mathrm{ℝ}$, clearly labelling the minimum point with its coordinates.

(ii) Explain why the function $f$ 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 $\left(g\circ f\right)\left(x\right)$= 21.

The function $f$ is defined by $f\left(x\right)=\frac{x+2}{4x-1}$.

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

It is always true that the graphs of a function and its inverse will be reflections of each other in the line $y=x$.

(ii) Based on the answer to part (i), state what else will be true about the graphs of $f$ and $f^{-1}$.

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

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  $f\left(x\right)=\sqrt{5x^2-11x+6.05}$ .

It is given that the inverse function f ^-1 exists, and that the domain of f  is as large as possible,

suggest two possible domains for f and write down the corresponding ranges.

Find what the value of $f^{-1}\left(\sqrt{22.05}\right)$ would be for each of the domains suggested in part (a).

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 f 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 as $x$ gets large $f\left(x\right)$ tends towards the value −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 $h\left(x\right)=-4x^2+24x+8,x\in \mathrm{ℝ}$.

(i) Show that $-4{\left(x-3\right)}^2=-4x^2+24x-36$.

(ii) Hence show that $h\left(x\right)=-4{\left(x-3\right)}^2+44$.

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

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

Describe a single transformation that would map 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 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(x)

(ii) g ^-1(x).

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

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

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

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

(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 $\left[a,b\right]$ is as large as possible. 

Find the domain and range of $f^{-1}$