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

Ask by Lane Donnelly. in the United States
Mar 11,2025

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Answer

The simplified difference quotients are: a) 2 b) \( 2x + h \) c) \( \frac{-2}{x(x+h)} \)

Solution

### a) For \( f(x)=2x-3 \) 1. **Compute \( f(x+h) \):** \[ f(x+h)=2(x+h)-3=2x+2h-3 \] 2. **Compute the difference quotient:** \[ \frac{f(x+h)-f(x)}{h}=\frac{(2x+2h-3)-(2x-3)}{h}=\frac{2h}{h} \] 3. **Simplify:** \[ \frac{2h}{h}=2 \] --- ### 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. **Compute the difference \( g(x+h)-g(x) \):** \[ g(x+h)-g(x)=\left(x^{2}+2xh+h^{2}+3\right)-\left(x^{2}+3\right)=2xh+h^{2} \] 3. **Compute the difference quotient:** \[ \frac{g(x+h)-g(x)}{h}=\frac{2xh+h^{2}}{h} \] 4. **Simplify:** \[ \frac{2xh}{h}+ \frac{h^{2}}{h}=2x+h \] --- ### c) For \( h(x)=\frac{2}{x} \) 1. **Compute \( h(x+h) \):** \[ h(x+h)=\frac{2}{x+h} \] 2. **Compute the difference \( h(x+h)-h(x) \):** \[ h(x+h)-h(x)=\frac{2}{x+h}-\frac{2}{x} \] 3. **Combine the fractions:** \[ \frac{2}{x+h}-\frac{2}{x}=\frac{2x-2(x+h)}{x(x+h)}=\frac{2x-2x-2h}{x(x+h)}=\frac{-2h}{x(x+h)} \] 4. **Compute the difference quotient:** \[ \frac{h(x+h)-h(x)}{h}=\frac{\frac{-2h}{x(x+h)}}{h}=\frac{-2h}{h \cdot x(x+h)} \] 5. **Simplify (cancel \( h \) assuming \( h\neq 0 \)):** \[ \frac{-2}{x(x+h)} \] --- Thus, the simplified difference quotients are: a) \( 2 \) b) \( 2x+h \) c) \( \displaystyle \frac{-2}{x(x+h)} \)

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To compute and simplify the difference quotient for each function, we use the formula \( \frac{f(x + h) - f(x)}{h} \) where \( h \) is a small increment approaching zero. Let's go through each function step by step. **a)** For \( f(x) = 2x - 3 \): 1. Calculate \( f(x + h) = 2(x + h) - 3 = 2x + 2h - 3 \). 2. Now, compute \( f(x + h) - f(x) = (2x + 2h - 3) - (2x - 3) = 2h \). 3. So, the difference quotient becomes \( \frac{f(x + h) - f(x)}{h} = \frac{2h}{h} = 2 \). Thus, the simplified difference quotient for \( f(x) \) is **2**. --- **b)** For \( g(x) = x^2 + 3 \): 1. Calculate \( g(x + h) = (x + h)^2 + 3 = x^2 + 2xh + h^2 + 3 \). 2. Now, compute \( g(x + h) - g(x) = (x^2 + 2xh + h^2 + 3) - (x^2 + 3) = 2xh + h^2 \). 3. So, the difference quotient becomes \( \frac{g(x + h) - g(x)}{h} = \frac{2xh + h^2}{h} = 2x + h \). Thus, the simplified difference quotient for \( g(x) \) is **2x + h**. --- **c)** For \( h(x) = \frac{2}{x} \): 1. Calculate \( h(x + h) = \frac{2}{x + h} \). 2. Now, compute \( h(x + h) - h(x) = \frac{2}{x + h} - \frac{2}{x} \). 3. This can be combined: \[ \frac{2}{x + h} - \frac{2}{x} = \frac{2x - 2(x + h)}{x(x + h)} = \frac{-2h}{x(x + h)}. \] 4. So, the difference quotient becomes \( \frac{h(x + h) - h(x)}{h} = \frac{-2h}{h \cdot x(x + h)} = \frac{-2}{x(x + h)} \). Thus, the simplified difference quotient for \( h(x) \) is **\(-\frac{2}{x(x + h)}\)**.

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