Practice section 1 of 2

For $f(x)=2x+1$ and $g(x)=x^2-3$, both on $\mathbb R$, can you find $gf(2)$? Next let $f(x)=x^2$ on $\{3,4,5\}$ and $g(x)=x+2$ on $\{2,3\}$. What is the rule of $fg$? State its domain and range. Finally, state the condition for $gf$ to exist.

Let $f$ be a one-to-one function defined by $f:x\mapsto2x+1$, $x\in\mathbb R$, $x>0$. Define (a) the inverse function $f^{-1}$, (b) the function $f^{-1}f$, and (c) the function $ff^{-1}$. Is $f^{-1}f=ff^{-1}$?

A function $f$ is said to be self-inverse if $f(x)=f^{-1}(x)$ for all $x$ in its domain. The functions are defined by $f:x\mapsto\dfrac{7-3x}{3-x}$, $x\in\mathbb R$, $x\ne3$, and $g:x\mapsto|(2-x)(1+x)|$, $x\in\mathbb R$, $x\le-1$. (i) Explain why $f^{-1}$ exists and show that $f$ is self-inverse. Hence, or otherwise, evaluate $f^{2003}(5)$. (ii) Find an expression for $g^{-1}(x)$. (iii) Sketch $g$ and $g^{-1}$ on the same diagram, illustrating their relationship and labelling axial intercepts, if any. Write down the set of values of $x$ satisfying $gg^{-1}(x)=x$. (iv) Show that $fg^{-1}$ exists and find its exact range.

The functions $f$ and $g$ are defined as

$f(x)=1-x^2$, $x\in\mathbb R$, $-1\leq x\leq1$,

$g(x)=\sqrt{x}+2$, $x\in\mathbb R$, $x\geq0$.

(i) Show that the function $gf$ exists.

(ii) Define the function $gf$ and sketch the graph of $gf$, stating the range of $gf$.

(iii) Explain why $fg$ does not exist.

Functions $f$, $g$ and $h$ are defined by

$f:x\mapsto|x^2-6x+5|$ for $x\in\mathbb R$, $2\leq x\leq6$,

$g:x\mapsto\ln(x^2)$ for $x\in\mathbb R$, $x\ne0$,

$h:x\mapsto\dfrac1{2-x}$ for $x\in\mathbb R$, $x\leq\dfrac32$.

(i) Give a reason why $f$ does not have an inverse.

(ii) State the longest possible interval in the domain of $f$, to which the restriction of $f$ would have an inverse. Hence, define $f^{-1}$ in similar form for function $f$ restricted to this interval.

(iii) Show that the composite function $gh$ exists.

(iv) Determine the exact range of function $gh$.

The function $f$ is defined by $f:x\mapsto\dfrac{ax}{bx-a}$ for $x\in\mathbb R$, $x\ne\dfrac ab$, where $a$ and $b$ are non-zero constants.

(i) Find $f^{-1}(x)$. Hence or otherwise find $f^2(x)$ and state the range of $f^2$.

(ii) The function $g$ is defined by $g:x\mapsto\dfrac1x$ for all real non-zero $x$.

State whether the composite function $fg$ exists, justifying your answer.

(iii) Solve the equation $f^{-1}(x)=x$.