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

To compute and simplify the difference quotient for each function, we use the formula: \[ \frac{f(x + h) - f(x)}{h} \] for a small increment \( h \). **a)** For \( f(x) = 2x - 3 \): 1. Compute \( f(x + h) \): \[ f(x + h) = 2(x + h) - 3 = 2x + 2h - 3 \] 2. Subtract \( f(x) \) from \( f(x + h) \): \[ f(x + h) - f(x) = (2x + 2h - 3) - (2x - 3) = 2h \] 3. Divide by \( h \): \[ \frac{2h}{h} = 2 \quad (h \neq 0) \] **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. Subtract \( g(x) \) from \( g(x + h) \): \[ g(x + h) - g(x) = (x^{2} + 2xh + h^{2} + 3) - (x^{2} + 3) = 2xh + h^{2} \] 3. Divide by \( h \): \[ \frac{2xh + h^{2}}{h} = 2x + h \quad (h \neq 0) \] **c)** For \( h(x) = \frac{2}{x} \): 1. Compute \( h(x + h) \): \[ h(x + h) = \frac{2}{x + h} \] 2. Subtract \( h(x) \) from \( h(x + h) \): \[ h(x + h) - h(x) = \frac{2}{x + h} - \frac{2}{x} \] To combine, find a common denominator: \[ = \frac{2x - 2(x + h)}{x(x + h)} = \frac{2x - 2x - 2h}{x(x + h)} = \frac{-2h}{x(x + h)} \] 3. Divide by \( h \): \[ \frac{-2h}{h \cdot x(x + h)} = \frac{-2}{x(x + h)} \quad (h \neq 0) \] So, the difference quotients are: - a) \( 2 \) - b) \( 2x + h \) - c) \( \frac{-2}{x(x + h)} \)