Further rational functions, odd/even and self-inverse functions, inequalities and modulus

Solve the inequality $|2x-3|<|x+1|$.

The diagram shows the graph of $y=|x+2|+|x-1|$ and the line $y=5$.

(a) Find the x-coordinates of the points where the graph meets the line $y=5$. [3]

(b) Hence solve $|x+2|+|x-1|<5$. [1]

(c) Solve $|x+2|+|x-1|=x+4$. [4]

Let $f(x)=\dfrac{x}{x^2+1}$, $x\in\mathbb{R}$, and $g(x)=x^2-|x|$, $x\in\mathbb{R}$.

(a) Show that $f$ is an odd function. [2]

(b) Determine whether $g$ is odd, even or neither. Justify your answer. [2]

(c) Hence, or otherwise, write down the value of $\displaystyle\int_{-5}^{5} f(x)\,\mathrm{d}x$. [1]

The function $f$ is defined by $f(x)=\dfrac{3x+5}{2x-3}$, $x\in\mathbb{R}$, $x\neq\tfrac32$.

(a) Show that $(f\circ f)(x)=x$. [3]

(b) Hence write down an expression for $f^{-1}(x)$. [1]

(c) State what the result in part (a) tells you about the symmetry of the graph of $y=f(x)$. [1]

Consider $f(x)=\dfrac{x^2+3x-2}{x-1}$, $x\in\mathbb{R}$, $x\neq1$.

(a) Write down the equation of the vertical asymptote of the graph of $y=f(x)$. [1]

(b) Find the equation of the oblique asymptote of the graph of $y=f(x)$. [2]

(c) Find the exact $x$-coordinates of the points where the graph of $y=f(x)$ meets the $x$-axis. [2]

(d) Write down the coordinates of the point where the graph of $y=f(x)$ meets the $y$-axis. [1]

Solve the inequality $3\sin x\geq x^2-1$, where $x\in\mathbb{R}$ is in radians.

Solve the inequality $|x^2-4|<3x$.

The diagram shows the graph of $y=f(x)$ for $-3\leq x\leq 4$. The graph consists of three straight line segments joining the points $P(-3,-1)$, $Q(-1,3)$, $R(2,-3)$ and $S(4,1)$.

(a) The graph of $y=|f(x)|$ is drawn. Write down the coordinates of the points to which $P$ and $R$ are mapped. [2]

(b) Find all the $x$-intercepts of the graph of $y=f(|x|)$, for $-4\leq x\leq 4$. [3]

(c) Find the number of solutions of the equation $|f(x)|=2$. [2]

Let $f(x)=x^2-4x+3$, $x\in\mathbb{R}$.

(a) Sketch the graph of $y=\dfrac{1}{f(x)}$, stating the equations of all asymptotes and the coordinates of any local maximum or minimum point and any axis intercepts. [4]

(b) For the graph of $y=[f(x)]^2$, write down the coordinates of
(i) the local minimum points;
(ii) the local maximum point;
(iii) the $y$-intercept. [3]

Consider the function $f(x)=\dfrac{2x-1}{x^2-x-2}$, $x\in\mathbb{R}$, $x\neq-1$, $x\neq2$.

(a) Write down the equations of the asymptotes of the graph of $y=f(x)$. [3]

(b) Show that $f'(x)=-\dfrac{2x^2-2x+5}{(x^2-x-2)^2}$. [2]

(c) Hence show that the graph of $y=f(x)$ has no stationary points. [2]

(d) Solve the inequality $f(x)\geq0$. [3]