# Composite and inverse functions

### Exercise

##### Question 1

Consider $f(x) = 4-x^{2}$ , $g(x)=\sqrt{x + 3}$ , $h(x) =\frac{1}{2x}$. Evaluate the following.
(a) $(f \circ g)(1)$
(b) $(g \circ h)(1)$
(c) $(f \circ g)(x)$
(d) $(g \circ h)(x)$
(e) $(h \circ g)(x)$
(f) $(f \circ g \circ h)(x)$

##### Question 2

Using the functions given in the previous question, explain why $(f \circ g)(-4)$ does not and cannot exist.

##### Question 3

Let $s(x)=\sqrt{x}$ and $t(x)=x^{2} + 2x + 1$. Evaluate $(s \circ t)(x)$.

##### Question 4

Show the following function pairs are inverses:
(a) $f(x) = x^{2}$ and $g(x) = \sqrt{x}$
(b) $f(x) = \frac{1}{x^{2}-1}$ and $g(x) =\sqrt{\frac{1}{x} + 1}$
(c) $f(x) = 2x-2$ and $g(x) = \frac{x}{2}+1$

##### Question 5

Find the inverses of the following functions.
(a) $y = 3x + 2$, (b) $y =\frac{1}{4-x}$
(c) $y = \frac{x+2}{x+5}$, (d) $y = x^{3}+ 1$

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