tssm topic 3

tssm topic 3
functions and relations - inverse functions, hybrid functions, modulus functions, composite functions and functional equations

when a function is stated but its domain is not specified, the function has an implied domain, which is the largest domain the function can take without resulting in a math error.

for example, the function $f(x)=\frac{1}{x}$ has the implied domain $x\in \mathbb{R}\backslash \{0\}$

1. solve for x

   1. $|x+5|=7$
   2. $-2|x-4|=24$

2. a function has rule $f(x)=5x^{\frac{1}{3}}-1,x\in \mathbb{R}$.

   1. the rule for the inverse function is:

   2. find the inverse, $f^{-1}$, of the function $f:\mathbb{R}\backslash \{\frac{-3}{5}\}\to \mathbb{R},f(x)=\frac{2x+1}{5x+3}$. state the domain and range of the inverse function.

   3. sketch the graph of $f:[3,\infty )\to \mathbb{R},f(x)=\sqrt{x-3}$

3. for the function $f(x)=2e^{3x}-4$ find its inverse function. sketch $f(x)$ and $f^{-1}(x)$ and state the domain and range of each

4. $f(x)=\{\begin{array}{ll}-2x+6& \text{if }\ge -2\\ x+2& \text{if }x<-2\end{array}$

   1. find $f(4)$
   2. sketch $f(x)$ and state its range

5. do these questions

   1. if $f(x)=x^2+4$, sketch $\frac{1}{f(x)}$
   2. if $f(x)=x^2+6x+8$, sketch $\frac{1}{f(x)}$

6. sketch the graph of $f(x)=2|x+4|+1$ and state the domain and range.

7. sketch the graph of $f(x)=|x^2=6|+1$ and state the domain and range.

8. sketch the graph of $f(x)=e^{|x|}$ and state the domain and range.

9. sketch the graph of $f(x)=x^2-5x$. hence sketch $f(|x|)$ and state the domain and range of $f(|x|)$.

10. for $f(x)=x^2$ and $g(x)=1-4x$, find

    1. $f(g(x))$
    2. $g(f(x))$
    3. $f(g(-1))$
    4. $g(f(-1))$

11. for $f(x)=e^{-x}$ and $g(x)=-x$, find the rules for:

    1. $(f+g)$
    2. $(fg)$

12. not available

13. prove the following functional equations

    1. $f(x+y)=f(x)f(y)$ for $f(x)=e^x$
    2. $f(xy)=f(x)+f(y)$ for $f(x)={\log }_2(x)$

14. if $\mathbf{A}=\left[\begin{array}{c}2\\ -1\end{array}\right]$, $\mathbf{B}=\left[\begin{array}{cc}1& 2\\ -2& 5\end{array}\right]$ and $\mathbf{A}=\left[\begin{array}{cc}2& 3\\ 1& 5\end{array}\right]$, find the following:

    1. $3\mathbf{A}$
    2. $2\mathbf{B}-\mathbf{C}$
    3. $\det (\mathbf{C})$
    4. ${\mathbf{B}}^{-1}$

15. state the implied domain for the following functions

    1. $f(x)=\ln (x-3)$
    2. $f(x)=\frac{1}{2x-1}$
    3. $f(x)=\sqrt{3x-4}$