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) \((g \circ h)(3)\)