Modulus Functions & Further Transformations (DP IB Analysis & Approaches (AA): HL): Exam Questions

Consider the function $f\left(x\right)=x^2-4\left|x\right|-5,x\in \mathrm{ℝ}.$ 

Solve $f\left(x\right)=0.$

Sketch the graph of $f$.
Clearly indicate the intersections with the coordinate axes and any turning points.

Given that

$f\left(x\right)=e^x,x\in \mathrm{ℝ},$ 

sketch on separate axes the graphs of

(i) $y=f\left(\right|x-1\left|\right)$

(ii) $y=\left|f\right(x)-1|$

(iii) $y=f(-|x\left|\right).$

Show any intercepts with the axes, label any local maximum and minimum points and give the equations of any asymptotes. Leave numbers in terms of $e$ where appropriate.

Consider the function $f$ defined by $f\left(x\right)=4x^2-8x-5$.

Sketch the graph of $y=\left|f\left(x\right)\right|$. Clearly indicate any intercepts with the axes and any turning points.

Sketch the graph of $y={\left[f\left(x\right)\right]}^2$. Clearly indicate any intercepts with the axes and any turning points.

Sketch the curve $y=\frac{4}{x+3}$ and the line $y=3-x$ on the same diagram, clearly indicating any $x$- and  $y$- axes intercepts as well as any asymptotes.

Hence find the exact solutions to the equation

$3-x=\left|\frac{4}{x+3}\right|.$

The graph of $f$ has two asymptotes with equation $x=0.25$ and $y=0.5$ as shown below.
The graph passes through the points $\mathrm{A}(0,6)$ and $\mathrm{B}(3,0)$.

Sketch the graph of $y=\frac{1}{f\left(x\right)}$.
Clearly indicate the points where the graph intersects the axes or has a discontinuity and state the equations of any asymptotes.

Consider the function  defined by $f\left(x\right)=\ln \left(k-x\right)$ where $f$ has the largest possible valid domain and $k$ is a positive constant such that $k>1$.

Sketch the graph of $y=f\left(x\right)$. Give the equations of any asymptotes and any intercepts with the axes in terms of $k$. Clearly state the domain and range of $f$.

The function $g$ is defined by $g\left(x\right)=f\left(\left|x\right|\right)$. The range of $g$ is $g\left(x\right)\le 1.$

(i) Find the exact value of $k$.

(ii) State the domain of $g$ .

(iii) Sketch the graph of $g$.

(c) Given that $\left|g\left(x\right)\right|=p$  has exactly two distinct real solutions find the range of values of $p$.

Find the set of values of $x$ which satisfy the inequality

$\left|2x^2-13x+15\right|>3x-15$

By considering the inverse of an appropriate function, sketch the graph $y=\sqrt{x}$ .

The function $f$ is defined by $f\left(x\right)=\sqrt{16-2x}$ and its domain is the largest possible set of real values. 

(i)     State the domain and range of $f$ . 

(ii)    Sketch the graph of $y=f\left(x\right)$.
Clearly label the points where the graph intersects the axes.

On separate sets of axes, sketch the graphs of:

(i) $y=f\left(\left|x\right|\right)$

(ii) $y={\left[f\left(x\right)\right]}^2$.  

For each graph, define the domain and range and clearly label the points where the graph intersects the axes.

By considering graph transformations of an appropriate quadratic function, sketch the graph of $y={\left(x+4\right)}^2-2\left|x+4\right|-3$ . Clearly indicate any $x$-intercepts and any $y$-intercepts.

Hence, find the values of $k$ such that the equation $(x+4{)}^2-2|x+4|-3=k$ has exactly 4 distinct real solutions.

A function $f$ is defined by $f\left(x\right)=(x-1)(x+2)(x-3),x\in \mathrm{ℝ}.$ 

(i) Sketch the graph of $y=\left|f\left(x\right)\right|$.

(ii) Hence find the set of values of $x$ for which $f\left(x\right)<\left|f\left(x\right)\right|$

Solve the equation $f\left(x\right)=|x^2-4x+3|.$

A function $f$ is defined by $f\left(x\right)=x^2-8x+15,x\in \mathrm{ℝ}.$

Sketch the graph of $y=\frac{1}{f\left(x\right)}$.  
Clearly indicate any intercepts with the coordinate axes and state the equations of any asymptotes.
Find the coordinates of any turning points.

By sketching the graphs of  $y=f\left(x\right)$ and $y=\frac{1}{f\left(x\right)}$ on the same axes, find the values of $x$  for which  $f\left(x\right)\le \frac{1}{f\left(x\right)}$.

Consider the function $f\left(x\right)=\frac{\left(x+3\right)\left(x-4\right)}{x-5},x\in \mathrm{ℝ},x\ne 5.$The graph of $f$  is shown below.

Find the equation of the oblique asymptote of the graph of $f$ .

Sketch the graph of $y=\left|f\left(x\right)\right|$.

Clearly indicate the points where the graph crosses the coordinates axes and state the equations of the asymptotes.

Sketch the graph of $y=f\left(\left|x\right|\right)$.
Clearly indicate the points where the graph crosses the coordinates axes and state the equations of the asymptotes.

The graph of a function $f$ is shown below. The equations of the asymptotes are $x=0$, $y=-1$ and $y=1.$

The point A(1, 2)  lies on the graph.

On separate sets of axes, sketch the graphs defined below.

For each sketch, clearly label any asymptotes or discontinuities and clearly show the coordinates of the point where A gets mapped to.

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

$y={\left[f\left(x\right)-2\right]}^2$.

$y=\frac{1}{{\left[f\left(x\right)\right]}^2}$

Let $f$ be the function defined by $f\left(x\right)=(x-1)(x^2+x-26).$ 

Sketch the graph of $y=\left|f\left(x\right)\right|$.
Give the exact coordinates of the $x$-intercepts.

Use differentiation to find the set of values of $k$ such that there are 4 points of intersection between the graph of $y=\left|f\left(x\right)\right|$ and the line $y=k$.

The graph of the function $f\left(x\right)=a|x+p|+q$is shown below, where $a,p,q\in \mathrm{ℝ}$.
The graph has a local maximum at the point A(3, 5)  and intersects the $y$-axis at (0,-7).

Find the values of $a,p$ and $q.$Hence solve $f\left(x\right)=0$.

Find the solutions to the equation $f\left(x\right)=\left|7-2x\right|$.

Consider the function defined by $f\left(x\right)=x\left|x\right|,x\in \mathrm{ℝ}.$

Sketch the graph of $f$.

By first sketching the graph of $f^{-1}$  on the same set of axes as the graph of $f$, solve $f\left(x\right)=f^{-1}\left(x\right).$

Given that $k$ is a real constant, find an expression for $f^{-1}\left(k\right)$ in the case when:

(i) $k\ge 0,$

(ii) $k<0$.

The graph of a function $f$ is shown below.

On separate sets of axes sketch the following graphs clearly showing any key points.

$y=\left|f\right(x\left)\right|-f\left(x\right).$

$y=f\left(x\right)\left|f\right(x\left)\right|$

$y=\frac{f\left(x\right)}{\left|f\left(x\right)\right|}$

 $f\left(x\right)=x^4-8x^3-40x^2+224x-240.$

Find the values of  such that $\left|f\left(\left|x\right|\right)\right|=k$ has six distinct, real solutions. Explain each stage of your solution in full.