Question

in problems 79 - 90, find the difference quotient of f; that is, find $\frac{f(x + h)-f(x)}{h}$, $h
eq0$, for each function. be sure to simplify.

1. $f(x)=4x + 3$
2. $f(x)=-3x + 1$
3. $f(x)=x^{2}-4$
4. $f(x)=3x^{2}+2$
5. $f(x)=x^{2}-x + 4$
6. $f(x)=3x^{2}-2x + 6$
7. $f(x)=\frac{1}{x^{2}}$
8. $f(x)=\frac{1}{x + 3}$
9. $f(x)=\frac{2x}{x + 3}$
10. $f(x)=\frac{5x}{x - 4}$
11. $f(x)=sqrt{x - 2}$ hint: rationalize the numerator.
12. $f(x)=sqrt{x + 1}$

Explanation:

Step1: Encontrar \(f(x + h)\) para \(f(x)=4x + 3\)

Sustituir \(x\) por \(x + h\) en \(f(x)\): \(f(x + h)=4(x + h)+3=4x+4h + 3\)

Step2: Calcular la diferencia cuociente

\(\frac{f(x + h)-f(x)}{h}=\frac{(4x + 4h+3)-(4x + 3)}{h}\)

Step3: Simplificar la expresión

\(\frac{4x + 4h+3 - 4x - 3}{h}=\frac{4h}{h}=4\)

Answer:

4