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. Express the given function \(H\) as a composition of two functions \(f\) and \(g\) such that \( H(x)=(f \circ g)(x) \).      (i)  \( H(x) = \sqrt{2x+1} \)        (ii)  \( H(x)=\frac{3}{|x^2-1|}\)      solution 

8.  Find the function \(f(x)\) and \(g(x)\) so that \( (f \circ g )(x) = (3x-4)^5\).   solution