10 Compute and simplify the difference quotient for each function given. a) $$ f(x)=2 x-3 $$ b) $$ g(x)=x^{2}+3 $$ c) $$ h(x)=\frac{2}{x} $$

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### a) For $$ f(x)=2x-3 $$
1. Compute $$ f(x+h) $$:
   $$
   f(x+h) = 2(x+h)-3 = 2x+2h-3
   $$
2. Form the difference quotient:
   $$
   \frac{f(x+h)-f(x)}{h}=\frac{[2x+2h-3] - [2x-3]}{h}
   $$
3. Simplify the numerator:
   $$
   2x+2h-3 -2x+3 = 2h
   $$
4. Divide by $$ h $$:
   $$
   \frac{2h}{h}=2
   $$

### b) For $$ g(x)=x^{2}+3 $$
1. Compute $$ g(x+h) $$:
   $$
   g(x+h)= (x+h)^2 + 3 = x^2+2xh+h^2+3
   $$
2. Form the difference quotient:
   $$
   \frac{g(x+h)-g(x)}{h}=\frac{[x^2+2xh+h^2+3]-[x^2+3]}{h}
   $$
3. Simplify the numerator:
   $$
   x^2+2xh+h^2+3-x^2-3 = 2xh+h^2
   $$
4. Divide by $$ h $$:
   $$
   \frac{2xh+h^2}{h} = 2x+h
   $$

### c) For $$ h(x)=\frac{2}{x} $$
1. Compute $$ h(x+h) $$:
   $$
   h(x+h)= \frac{2}{x+h}
   $$
2. Form the difference quotient:
   $$
   \frac{h(x+h)-h(x)}{h}= \frac{\frac{2}{x+h}-\frac{2}{x}}{h}
   $$
3. Combine the fractions in the numerator:
   $$
   \frac{2}{x+h}-\frac{2}{x} = \frac{2x-2(x+h)}{x(x+h)} = \frac{2x-2x-2h}{x(x+h)} = \frac{-2h}{x(x+h)}
   $$
4. Divide by $$ h $$ and simplify:
   $$
   \frac{\frac{-2h}{x(x+h)}}{h} = \frac{-2h}{h\, x(x+h)} = \frac{-2}{x(x+h)}
   $$
### a) For $$ f(x)=2x-3 $$ $$ \frac{f(x+h)-f(x)}{h} = 2 $$ ### b) For $$ g(x)=x^{2}+3 $$ $$ \frac{g(x+h)-g(x)}{h} = 2x + h $$ ### c) For $$ h(x)=\frac{2}{x} $$ $$ \frac{h(x+h)-h(x)}{h} = \frac{-2}{x(x+h)} $$

10 Compute and simplify the difference quotient for each function given. a) \( f(x)=2 x-3 \) b) \( g(x)=x^{2}+3 \) c) \( h(x)=\frac{2}{x} \)

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