Further Functions & Graphs (DP IB Analysis & Approaches (AA): SL): Exam Questions

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

For the graph of $f$, find the:

(i) $x$-intercept

(ii) $y$-intercept.

For the graph of $f$, write down the equation of any asymptotes.

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

Write down the coordinates of P and Q.

Find the distance of PQ.

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

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

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

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

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

Find the value of $p.$

For the graph of $f$ write down the equation of any asymptotes.

Find the range of $f$.

The line  intersects the graph of $f$ when $x=1$ and when $x=4.$

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

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

Find the value of $a$. Give your answer as a fraction.

Find the range of f.

Find the value of $f^{-1}\left(2\right)$. Give your answer as a fraction.

Let $f\left(x\right)=\frac{4}{6-x}$, for $x\ne 6$.

For the graph of f, find 

(i) the x-intercept

(ii) the y-intercept

(iii) The equation of the vertical asymptote.

Let $g\left(x\right)=-\frac{x}{4}$, for $x\in \mathrm{ℝ}$. The graphs of f and g intersect at points A and B.

Find the coordinates of A and B.

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).

The number of bacteria, n, in a dish, after t minutes is given by $n=5231e^{0.12t}.$

Find the initial amount of bacteria.

Find the amount of bacteria after 12 minutes. Give your answer in the form $a\times {10}^k$, where $1\le a<10,k\in \mathrm{\mathbb{Z}}$

Find the value of t when $n=2.7\times {10}^4$.

Let $f\left(x\right)=e^{-x}+1$and $g\left(x\right)=2x-m$, for$x\in \mathrm{ℝ}$, where m is a constant.

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

Given that $\underset{x\to \infty }{\lim }\left(g\circ f\right)\left(x\right)=-1$ , find the value of m.

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

Find the value of $b$.

Find the range of $f$ .

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

Find the equation of the line passing through points A and B. Give your answer in the form $y=mx+c.$

Find the distance of the line 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 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 can be defined by $g\left(x\right)=6{\left(0.8\right)}^{-x}-1$, for $-10\le x\le 10$.

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

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

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

Write down the domain and range of the logarithmic function$y={\log }_{b}x$ , where $b>0$and $b\ne 1$.

Given that ${\log }_{y^2}x=16{\log }_x\left(y^2\right)$, find all the expressions for x in terms of y.

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 following diagram shows part of the graph of  f  which crosses the x-axis at point A, with coordinates ( a, 0).  The line L is the tangent to the graph of   f   at the point B. 

Find the exact value of a.

The x-coordinate of B is 10. The y-coordinate of B can be written in the form $p\ln q$ , where $p,q\in \mathrm{\mathbb{Z}}.$

Find the value of p and the value of q.

The gradient of L is $\frac{5}{9}$ . The equation of L can be written in the form 

$y=\frac{5}{9}x-u(v-\ln w).$

Find the values of u, v and w.

A population of endangered birds, P, can be modelled by the equation

$P_t=P_0{e}^{kt},$

where P₀ is the initial population and t is measured in years. 

After three years, it is estimated that $\frac{P_3}{P_0}=0.87$.

Find the value of k and interpret its meaning.

Find the least number of whole years for which $\frac{P_t}{P_0}<0.45.$

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

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

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.

Calculate the intensity of light 6.2 m below the surface.

Let $f\left(x\right)=\frac{6-8x}{cx-12}$ , for $x\ne \frac{12}{c}$ , where $c\ne 0$.

The line x=2 is a vertical asymptote to the graph of $y=f\left(x\right).$

(i) Find the value of c.

(ii) Write down the equation of the horizontal asymptote to the graph of $y=f\left(x\right).$

The line$y=h$, where $h\in \mathrm{ℝ}$, intersects the graph of $y=\left|f\left(x\right)\right|$ at exactly one point. Find possible values of $h$.