Chapter 6

Find the inverse of each of the following functions: (a) \(f(x)=4 x+7\) (b) \(f(x)=x\) (c) \(f(x)=-23 x\) (d) \(f(x)=\frac{1}{x+1}\)

Given the function \(f(t)=2 t^{2}+4\) find (a) \(f(x)\) (b) \(f(2 x)\) (c) \(f(-x)\) (d) \(f(4 x+2)\) (e) \(f(3 t+5)(\mathrm{f}) f(\lambda)(\mathrm{g}) f(t-\lambda)\) (h) \(f\left(\frac{t}{\alpha}\right)\)

If \(f(x)=x+6\) and \(g(x)=x^{2}-5\) find (a) \(f(g(0))\), (b) \(g(f(0))\), (c) \(g(g(2))\), (d) \(f(g(7))\).

Study graphs of \(y=3 x-2\) and \(y=-7 x+1\). Are these continuous functions?

If \(f(x)=\frac{x-3}{x+1}\) and \(g(x)=\frac{1}{x}\) find \(g(f(x))\).

Given \(g(x)=3 x^{2}-7\) find (a) \(g(3 t)\) (c) \(g(6 t-4)\) (d) \(g(4 x+9)\)

Explain the meaning of the terms 'domain' and 'range' when applied to functions.

Draw a graph of the function $$ f(x)= \begin{cases}2 x+1 & x<3 \\ 5 & x=3 \\ 6 & x>3\end{cases} $$ Find (a) \(\lim _{x \rightarrow 0^{+}} f(x)\) (b) \(\lim _{x \rightarrow 0}-f(x)\) (c) \(\lim _{x \rightarrow 0} f(x)\) (d) \(\lim _{x \rightarrow 3^{+}} f(x)\) (e) \(\lim _{x \rightarrow 3^{-}} f(x)\) (f) \(\lim _{x \rightarrow 3} f(x)\)

A periodic function has period \(0.01\) seconds. How many times will the pattern in the graph repeat over an interval of 10 seconds?

Plot a graph of the following functions. In each case state the domain and the range of the function. (a) \(f(x)=3 x+2,-2 \leq x \leq 5\) (b) \(g(x)=x^{2}+4,-2 \leq x \leq 3\) (c) \(p(t)=2 t^{2}+8,-2 \leq t \leq 4\) (d) \(f(t)=6-t^{2}, 1 \leq t \leq 5\)