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

Question

UpStudy AI · Accepted Answer

### 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 $$ h\neq 0 $$):**

$$
   \frac{-2}{x(x+h)}
   $$

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Thus, the simplified difference quotients are:

a) $$ 2 $$

b) $$ 2x+h $$

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

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