2.8 Modulus Functions & Inequalities

A function `f` is defined by `f(x)={2x-1}/{x+1}`, where `x in RR, x ne 1`.

(a) The graph of `y=f(x)` has a vertical asymptote and a horizontal asymptote. 

Write down the equation of

(i) the vertical asymptote; 

(ii) the horizontal asymptote. 

(b) On the set of axes below, sketch the graph of `y=f(x)`. 

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.

(c) Hence, solve the inequality `0 < {2x-1}/{x+1} < 2`.

(d) Solve the inequality `0 < {2|x|-1}/{|x|+1} < 2`.

(a)

(i) `x=-1`

(ii) `y=2`

(b)

rational function shape with two branches in opposite quadrants, with two correctly positioned asymptotes and asymptotic behaviour shown

axes intercepts clearly shown at `x=1/2` and `y=-1`

(c) `x > 1/2`

(d) attempts to solve `2|x|-1=0` then `x < -1/2` or `x > 1/2`

The function `f` is defined by `f(x) = cos^2 x - 3 sin^2 x, 0 <= x <= pi`.

(a) Find the roots of the equation `f(x)=0`.

(b) 

(i) Find `f'(x)`.

(ii) Hence find the coordinates of the points on the graph of `y=f(x)` where `f'(x)=0`.

(c) Sketch the graph of `y=|f(x)|`, clearly showing the coordinates of any points where `f'(x)=0` and any points where the graph meets the coordinate axes. 

(d) Hence or otherwise, solve the inequality `|f(x)| > 1`.

(a) `cos^2 x - 3 sin^2 x = 0`

valid attempt to reduce equation to one involving one trigonometric function

`sin^2x/cos^2x=1/3`

correct equation

`tan^2x=1/3`

`tan x = +- 1/sqrt3`

`x = pi/6, x={5pi}/6`

(b) 

(i) attempt to use the chain rule (may be evidenced by at least one `cos x sin x` term)

`f'(x)= -2cos x sin x - 6 sin x cos x`

`(= -8 sin x cos x = -4 sin 2x)`

(ii) valid attempt to solve their `f'(x)=0`

at least 2 correct `x`-coordinates (may be seen in coordinates)

`x=0,x=pi/2,x=pi`

correct coordinates (may be seen in graph for part (c))

`(0,1), (pi,1), (pi/2, -3)`

(c) 

attempt to reflect the negative part of the graph of `f` in the 𝑥-axis

endpoints have coordinates `(0,1), (pi,1)`

smooth maximum at `(pi/2,3)`

sharp points (cusps) at `x`-intercepts `pi/6, {5pi}/6`

(d) considers points of intersection of `y=|f(x)|` and `y=1` on graph or algebraically 

`-(cos^2x-3sin^2x)=1`

`tan^2x=1`

`x=pi/4,{3pi}/4`

For `|f(x)| > 1`

`pi/4 < x < {3pi}/4`

Let `f(x)=4 cos (x/2)+1`, for `0 <= x <= 6pi`. Find the values of `x` for which `f(x) > 2sqrt2 + 1`.

`4cos(x/2)+1 > 2 sqrt 2 + 1`

correct working

e.g. `4cos(x/2)=2 sqrt 2, cos (x/2) > sqrt 2 /2`

Find the value of `x` for which `|5-3x| <= |x+1|`.

We obtain A = (1, 2) and B = (3, 4)

Therefore, `1 <= x <= 3`

Solve the inequality `x^2-4+3/x < 0`.

`x^2 - 4 + 3/x < 0`

`=> -2.30 < x < 0` or `1 < x < 1.30`

Solve the inequality `|x-2| >= |2x+1|`.

The graphs of `y=|x-2|` and `y=|2x+1|` meet where 

`(x-2)=(2x+1) => x=-3`

`(x-2)=-(2x+1) => x=1/3`

Test any value, e.g. `x=0` satisfies inequality

so `x in [-3,1/3]`

`m(x+1) <= x^2 => x^2 - mx - m >= 0`

Hence `Delta = b^2 - 4ac <= 0`

`=> m^2 + 4m <= 0`

Solve `| ln(x+3)|=1`. Give your answers in exact form. 

Finding two equations

Correct equations `ln(x+3)=1, ln(x+3)=-1`

`ln(x+3)=1 => x = e-3`

`ln(x+3)=-1 => x=1/e -3 \ (=e^-1 -3)`

Determine the values of `x` that satisfy the following inequalities

(a) `{|x|+2}/{|x|-3} < 4`

(b) `{x e^x}/(x^2 - 1) >= 1`

(a) `x < -14/3 \ \ \ \ \ -3 < x < 3 \ \ \ \ \ x > 14/3`

(b) `−1 < x < −0.800` or `x > 1` (accept `-1 < x <= -0.800`)

The graph of the function `f(x) = 2cos(4x)-1`, where `0^@ <= x <= 90^@`, is shown on the diagram below. 

(a) On the diagram draw the graph of the function `g(x)=sin(2x)-2`, for `0^@ <= x <= 90^@`.

(b) Write down the number of solutions to the equation `f(x)=g(x)`, for `0^@ <= x <= 90^@`.

(c) Write down one value of `x` for which `f(x) > g(x)`, for `0^@ <= x <= 90^@`.

`f(x) < g(x)` in the interval `a < x < b`.

(d) Use your graphic display calculator to find the value of 

(i) `a`;

(ii) `b`.

(a)

(b) 2

(c) the value given must be between `0^@ <= x < 24.3^@` or `65.7^@ < x <= 90^@`

(d) 

(i) `a= 24.3^@`

(ii) `b=65.7^@`