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 can use the formula: \[ \frac{f(x+h) - f(x)}{h} \] for the function \( f(x) \) and similarly for \( g(x) \). **a) For \( f(x) = 2x - 3 \):** 1. Calculate \( f(x + h) \): \[ f(x + h) = 2(x + h) - 3 = 2x + 2h - 3 \] 2. Now plug this into the difference quotient: \[ \frac{f(x+h) - f(x)}{h} = \frac{(2x + 2h - 3) - (2x - 3)}{h} \] This simplifies to: \[ \frac{2x + 2h - 3 - 2x + 3}{h} = \frac{2h}{h} = 2 \quad \text{(for } h \neq 0\text{)} \] So, the difference quotient for \( f(x) \) is simply \( 2 \). **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. Now plug this into the difference quotient: \[ \frac{g(x+h) - g(x)}{h} = \frac{(x^2 + 2xh + h^2 + 3) - (x^2 + 3)}{h} \] This simplifies to: \[ \frac{x^2 + 2xh + h^2 + 3 - x^2 - 3}{h} = \frac{2xh + h^2}{h} \] Dividing each term by \( h \) (again, for \( h \neq 0 \)): \[ 2x + h \] So, the difference quotient for \( g(x) \) is \( 2x + h \). In summary: - The difference quotient for \( f(x) = 2x - 3 \) is **2**. - The difference quotient for \( g(x) = x^2 + 3 \) is **\( 2x + h \)**.