Further Functions & Graphs (DP IB Applications & Interpretation (AI): SL): Exam Questions

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 $f^{-1}$.

Consider the functions $f\left(x\right)=-x^5+2020$ and $g\left(x\right)=\frac{1}{\sqrt{{\left(1-x\right)}^3}}-2$.

Find the coordinates of the $y$-intercepts for the graph of

(i) $f$

(ii) $g$.

Find the coordinates of the $x$-intercepts for the graph of

(i) $f$

(ii) $g.$

For the graph of $g$, find the equation of

(i) the vertical asymptote

(ii) the horizontal asymptote.

Consider the functions $f\left(x\right)=x^{-4}-2021$ and $g\left(x\right)=2-\sqrt{x-1}$. Find the maximum possible domain and range of $g.$

For the graph of $f$, find the equation of

(i) the vertical asymptote

(ii) the horizontal asymptote.

Find the coordinates of the $x$-intercepts for the graph of

(i) $f$

(ii) $g.$

Consider the functions $f\left(x\right)=-x^2-x+6$ and $g\left(x\right)=(2x+1{)}^2-9$.

Sketch the graphs of the functions $f\left(x\right)$ and $g\left(x\right)$ and label the coordinates of the vertices for both functions.

Find the coordinates for the points of intersection of $f\left(x\right)$ and $g\left(x\right)$.

Find the $x$-intercepts of $f\left(x\right)$ and $g\left(x\right)$.

The diagram below shows part of the graph of the function $f\left(x\right)=-x^2+bx+c$, where $b$ and $c$ are both integers. Points P(-2, 0) and R(6, 0) represent the $x$-intercepts, point Q(0, 12) represents the $\mathrm{y}$-intercept, point V represents the vertex of the graph of $f$ and O represents the origin (0, 0).

Write down the value of $c$.

Find the value of $b$ and write down $f\left(x\right).$

Write down the coordinates of $\mathrm{V}$.

The function $g\left(x\right)=a{x}^2+bx+c$ intercepts the $y$-axis at $-16$, has an $x$-intercept when $x=-4$ and can be obtained by an appropriate translation of the graph $y=2x^2$.

(i) Find the values of $a,b$ and $c$.

(ii) Write down $g\left(x\right)$

Find the other $x$-intercept of $g\left(x\right).$

Write down the coordinates of the vertex of $g\left(x\right)$.

The diagram below shows the graph of the function $f\left(x\right)=2\sin \left(2x\right)$ for $0°\le x\le 360°$.

State the amplitude of $f\left(x\right)$.

Calculate the period of $f\left(x\right)$.

Find the possible values of $x$ when $f\left(x\right)=-1.$

The diagram below shows part of the graph of the function $f\left(x\right)=x^3-x^2-4x+1.$

Points $A,C,D$ and $F$ represent where the graph of $f$ intersects the coordinate axes, write down the coordinates for

(i) $A$

(ii) $C$

(iii) $D$

(iv) $F$.

Points $B$ and $E$ represent the local maximum and minimum respectively for $f\left(x\right)$, write down the coordinates for

(i) $B$

(ii) $E$.

The diagram below shows part of the graph of the function $f\left(x\right)=2^x-3.$

Find

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

(ii) $x$when $f\left(x\right)=-1$.

The point P represents the $y$-intercept of $f\left(x\right).$Write down the coordinates of $P$.

The point Q represents the $x$-intercept of $f\left(x\right)$. Write down the coordinates of $Q$.

Draw the line $y=-3$ on the graph above.

Write down the number of solutions to the equation $f\left(x\right)=-3.$

The graph of a quadratic function has equation $y=\frac{1}{4}{x}^2+bx+c$, where $b,c\in \mathrm{\mathbb{Z}}$, and the axis of symmetry is $x=-4$.

Draw the axis of symmetry on the grid above.

The graph of the quadratic function intersects the $x$-axis at the points $\mathrm{A}\left(-6,0\right)$ and B.

(i) Write down the coordinates of B.

(ii) Find the values of $b$ and $c$.

(i) Mark and label A and B on the grid above.

(ii) Write down the coordinates of the vertex, V, and label it on the grid above.

(iii) Write down the coordinates of the $y$-intercept, C, and label it on the grid above.

(iv) Draw the graph of the quadratic function on the grid above.

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

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

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

A function is defined by $f\left(x\right)=4-\frac{12}{5x+9}$, for $x\in \mathrm{ℝ}$, $x\ne a$.

Write down the value of $a$.

Find the range of $f$.

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

Let $f\left(x\right)=\frac{7}{2\left(x-7\right)}-5$, for $x\ne 7$.

Find the coordinates of the point where the graph of $f$ crosses

(i) the $x$-axis

(ii) the $y$-axis.

Write down the equation of

(i) the vertical asymptote of the graph of $f$

(ii) the horizontal asymptote of the graph of $f$.

Let $g\left(x\right)=2\left(1-2x\right)$, for $x\in \mathrm{ℝ}$. The graphs of $f$ and $g$ intersect at the points $\mathrm{P}$ and $\mathrm{Q}$.

Write down the coordinates of $\mathrm{P}$ and of $\mathrm{Q}$.

Find $\mathrm{PQ}$.

Let $f\left(x\right)=x^2-3x+2$. The diagram below shows part of the graph of $f$.

Another function is defined by $g\left(x\right)=2-x.$

Sketch the graph of $g$ on the axes above.

The graph of $f$ and $g$ intersect at points A and B.

Find the coordinates of A and B and label them on the diagram above.

Find the length of the line AB.

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 $k$.

Consider the function $f\left(x\right)=a{\left(0.75\right)}^x+b$, where $a$ and $b$ are constants. The graph of $f$ passes through the points $\left(0,18\right)$ and $\left(2,11\right)$ and is shown below.

Write down two equations relating $a$ and $b$.

Find the value of $a$ and the value of $b$.

Write down the equation of the horizontal asymptote of the graph of $f$.

The function $f\left(x\right)=a{x}^2+bx+c$ intercepts the $y$-axis at $-12$ and has an $x$-intercept at $x=3.$ The function can be obtained by an appropriate shift of the graph $y=-4x^2$.

Find the values of $a,b$ and $c$.

Find the other $x$-intercept of $f\left(x\right)$.

Determine the coordinates of the maximum value of $f\left(x\right)$ .

A function is defined by $f\left(x\right)=\frac{1}{{\left(x-3\right)}^2}+2$, for $x\ne p$.

Find the value of $p$.

Write down the equation of

(i) the vertical asymptote of the graph of $f$

(ii) the horizontal asymptote of the graph of $f$.

Write down the range of $f$.

The line $l$ intersects the graph of $f$ at the points where $x=1$ and $x=4$.

Find the equation of $l$. Give your answer in the form $ax+by+d=0$, where $a$, $b$ and $d$ are integers.

The diagram below shows the graph of the function $f\left(x\right)=8\cos \left(48x\right)-3$, for $-5\le x\le 5$.

For the graph of $f$, find the:

(i) $x$-intercepts

(ii) $y$-intercept.

Write down the range of $f$.

Three lines are drawn connecting the two local minimum points and the local maximum point forming a triangle. Calculate the area of the triangle.

Let  $f\left(x\right)=2-5\cos \left(30\left(3x-1\right)\right),$ for $x>0.$

The $n$th maximum point on the graph of $f$ has $x$ coordinate $x_n$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Given that $x_n=x_1+\left(n-1\right)d$, find $x_1$ and $d$.

Using sigma notation, write down an expression for $x_1+x_2+x_3+\cdots x_8$.

Find the value of the sum from part (b) (i). Give your answer to 2 decimal places.

The average fat-free mass, M, in kg, of footballers as a function of their age, $a$, in years, can be given by the logarithmic function:

$\mathrm{M}\left(a\right)=10\log \left(a-15\right)+50,16\le a\le 25.$

Calculate the average fat free mass of players aged:

(i) 16 years

(ii) 25 years.

Find an expression for a linear model using your answers to part (a) (i) and (ii).

Calculate the percentage error from using the linear model found in part (b) to approximate the average fat free mass of a player aged 20 years old.

The axes below shows the graph of the piecewise function, $f$

The gradient of the graph of $f$ is 0 for $a\le x<b$.

Find the values of $a$ and $b$.

Complete the following piecewise function for $f$

The function $f$ is a quadratic in the form $f\left(x\right)=a{x}^2+bx-2$, for $-10\le x\le 10$.

The graph of $f$ has $x$-intercepts $\left(\frac{1+\sqrt{5}}{2},0\right)$ and $\left(\frac{1-\sqrt{5}}{2},0\right)$.

Find the values of $a$ and $b$.

Another function is defined by $g\left(x\right)=6{\left(0.8\right)}^{-x}-1$, for $-10\le x\le 10$.

The graphs of $f$ and $g$ intersect at the points $\mathrm{A}$ and $\mathrm{B}$.

Find the coordinates of $\mathrm{A}$ and of $\mathrm{B}$.

Hence, solve $f\left(x\right)<g\left(x\right)$.

A function is defined by $f\left(x\right)={\mathrm{e}}^{x^2+bx+4}$. The graph of $f$ has an axis of symmetry $x=2$.

Find the value of $b$.

Write down the range of $f$.

Another function is defined by $g\left(x\right)=-\frac{x^2-25}{5}$. The graphs of $f$ and $g$ intersect at the points $\mathrm{A}$ and $\mathrm{B}$.

Find the equation of the line $\left(\mathrm{AB}\right)$. Give your answer in the form $y=mx+c$.

Find $\mathrm{AB}$.

Consider the function $f\left(x\right)=5-\log (6-4x).$ The line $l_1$ intersects the graph of $f$ at point A$(-1,y)$ and B$(x,5)$.

Find the value of $x$ and $y$.

Find the equation of $l_1$. Give your answer in the form $y=mx+c$, where $m$ and $c$ are fractions.

The line $l_2$ is the perpendicular bisector to [AB].

Find the equation of $l_2$. Give your answer in the form $ax+by+d=0$, where $a,b$ and $d$ are integers.

Let $f\left(x\right)=e^{2cos\left(30x\right)}+1,$ for $x>0.$

The $n$th minimum point on the graph of $f$ has $x$ coordinate $x_n$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Given that $x_n=x_1+(n-1)d$, find $x_1\mathrm{and}d$.

(i) Using sigma notation, write down an expression for $x_1+x_2+x_3+\cdots +x_{16}.$

(ii) Find the value of the sum from part (b) (i). Give your answer to the nearest integer.

Let $f\left(x\right)=2x^4-2x^3-4x^2+x+1$ where $x\in \mathrm{ℝ}$.

Solve the inequality $f\left(x\right)<0$.

For the graph of $f$, find the coordinates of the

(i) local maximum point.

(ii) local minimum points.

Write down the possible domains of $f$ for which $f$ has an inverse and explain why the domain must be restricted.

The diagram below shows part of the graph of the function $f\left(x\right)=5\cos (12x-30)+6$, where points A, B, C and D represent stationary points. The points A, B, C and D are joined together forming a quadrilateral ABDC.

Write down the coordinates for

(i) $\mathrm{A}$.

(ii) $\mathrm{B}$.

(iii) $\mathrm{C}.$

(iv) $\mathrm{D}.$

Find the area of the quadrilateral ABDC.

The intensity of light, I, is assumed to be 100% at the surface of the ocean and decreases with depth, $d$, and can be estimated by the function

$I\left(d\right)=k{(1.08)}^{-d}$

where $I$ is expressed as a percentage, $d$ is the depth below the surface, in metres, and $k$ is a constant.

Calculate the value of $k$.

State the domain and range of $I$.

On a particularly bright day the intensity of light 6.2 m below the surface is 75%.

Calculate the percentage error between the estimate made by the function and the actual intensity of light.

Graph the following piecewise function on the grid below

Find the values of $x$ when $f\left(x\right)=0.$