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

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**a) For $$ f(x)=2x-3 $$:**

1. Compute $$ f(x+h) $$:
   $$
   f(x+h)=2(x+h)-3=2x+2h-3.
   $$

2. Find the difference:
   $$
   f(x+h)-f(x)=(2x+2h-3)-(2x-3)=2h.
   $$

3. Divide by $$ h $$:
   $$
   \frac{f(x+h)-f(x)}{h}=\frac{2h}{h}=2 \quad (\text{provided } h\neq 0).
   $$

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**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. Find the difference:
   $$
   g(x+h)-g(x)=(x^2+2xh+h^2+3)-(x^2+3)=2xh+h^2.
   $$

3. Divide by $$ h $$:
   $$
   \frac{g(x+h)-g(x)}{h}=\frac{2xh+h^2}{h}=2x+h \quad (\text{provided } h\neq 0).
   $$

---

**c) For $$ h(x)=2 $$:**

1. Compute $$ h(x+h) $$:
   $$
   h(x+h)=2.
   $$

2. Find the difference:
   $$
   h(x+h)-h(x)=2-2=0.
   $$

3. Divide by $$ h $$:
   $$
   \frac{h(x+h)-h(x)}{h}=\frac{0}{h}=0.
   $$
**a) For $$ f(x)=2x-3 $$:** $$ \frac{f(x+h)-f(x)}{h} = 2 $$ **b) For $$ g(x)=x^2+3 $$:** $$ \frac{g(x+h)-g(x)}{h} = 2x + h $$ **c) For $$ h(x)=2 $$:** $$ \frac{h(x+h)-h(x)}{h} = 0 $$

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

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