Composite Functions

1. Let $f(x)= \sqrt{x+1}$ and $g(x)=2x-7$. Find (i)  $f + g$,  (ii) $f - g$,   (iii) $f \cdot g$, and (iv)  $\frac{f}{g}$ and state their domains.  solution

2. The functions  $f(x)= 2x-1$ and $g(x)=2x^2-5$ are given. Find each of the given values. (i)  $(f + g)(-1)$,     (ii) $(f - g)(0)$,   (iii) $(f \cdot g)(2)$, and (iv) $\left(\frac{f}{g}\right)(1)$.   solution

3. The functions  $f(x)= \frac{2}{x+1}$ and $g(x)=\frac{x}{x+1}$ are given. Find each of the given values. (i)  $(f + g)(0)$,     (ii) $(f - g)(3)$,   (iii) $(f \cdot g)(2)$,  (iv) $\left(\frac{f}{g}\right)(1)$ and (v) $\left(\frac{g}{f}\right)(-2)$.   solution

4.  Let $f(x)= 2x+1$ and $g(x)=2x^2-3$. Evaluate (i)  $(f \circ g)(-1)$, and (ii)  $(f \circ f)(0)$.  solution

5. The functions  $f(x)= \frac{2}{x-1}$ and $g(x)=\frac{4}{x}$ are given.  Evaluate  $f \circ g$ and find its domain.  solution

6. The functions  $f(x)= 3-2x$ and $g(x)=\sqrt{2x-3}$ are given.  Evaluate  $g \circ f$ and find its domain.  solution

7. Use the table to evaluate the composite functions.  solution

$x$	1	2	3	4
$f(x)$	2	3	4	7
$g(x)$	3	1	3	-10
$h(x)$	3	3	1	-8

(a) $(f \circ g)(2)$   (b) $(h \circ h)(2)$   (c) $(g \circ f)(3)$   (d) $(g \circ h)(2)$   (e) $(h \circ g)(1)$  (f) $(g \circ h)(3)$