Compute and simplify the difference quotient for each function given \( \begin{array}{ll}\text { a) } f(x)=2 x-3 \\ \text { b) } g(x)=x^{2}+3 & \text { (10b) } g(x)=x^{2}+3\end{array} \)

To compute and simplify the difference quotient for the given functions, we will use the formula for the difference quotient: \[ \frac{f(x+h) - f(x)}{h} \] where \( h \) is a small increment. We will apply this formula to both functions \( f(x) \) and \( g(x) \). ### a) For \( f(x) = 2x - 3 \) 1. **Calculate \( f(x+h) \)**: \[ f(x+h) = 2(x+h) - 3 = 2x + 2h - 3 \] 2. **Calculate the difference quotient**: \[ \frac{f(x+h) - f(x)}{h} = \frac{(2x + 2h - 3) - (2x - 3)}{h} \] 3. **Simplify the expression**: \[ = \frac{2x + 2h - 3 - 2x + 3}{h} = \frac{2h}{h} = 2 \] Thus, the simplified difference quotient for \( f(x) \) is \( 2 \). ### b) For \( g(x) = x^2 + 3 \) 1. **Calculate \( g(x+h) \)**: \[ g(x+h) = (x+h)^2 + 3 = x^2 + 2xh + h^2 + 3 \] 2. **Calculate the difference quotient**: \[ \frac{g(x+h) - g(x)}{h} = \frac{(x^2 + 2xh + h^2 + 3) - (x^2 + 3)}{h} \] 3. **Simplify the expression**: \[ = \frac{x^2 + 2xh + h^2 + 3 - x^2 - 3}{h} = \frac{2xh + h^2}{h} \] \[ = 2x + h \] Thus, the simplified difference quotient for \( g(x) \) is \( 2x + h \). ### Summary of Results - For \( f(x) = 2x - 3 \), the difference quotient simplifies to \( 2 \). - For \( g(x) = x^2 + 3 \), the difference quotient simplifies to \( 2x + h \).