### About Difference Quotient:

In some cases, we are asked to find the difference quotient of a given function. Additionally, there are specific measures we need to take in order to find the difference quotient with fractions and with square roots.

Test Objectives

- Demonstrate the ability to find the difference quotient
- Demonstrate the ability to rationalize the numerator

#1:

Instructions: find the difference quotient.

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

$$a)\hspace{.2em}f(x)=5x - 3$$

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#2:

Instructions: find the difference quotient.

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

$$a)\hspace{.2em}f(x)=3x^2 - 5x - 1$$

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#3:

Instructions: find the difference quotient.

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

$$a)\hspace{.2em}f(x)=x^2 - 25$$

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#4:

Instructions: find the difference quotient.

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

$$a)\hspace{.2em}f(x)=\frac{7}{x - 1}$$

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#5:

Instructions: find the difference quotient.

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

$$a)\hspace{.2em}f(x)=\sqrt{x - 4}$$

On the problem above, we need to take the extra step of rationalizing the numerator

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Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}5$$

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#2:

Solutions:

$$a)\hspace{.2em}6x + 3h - 5$$

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#3:

Solutions:

$$a)\hspace{.2em}2x + h$$

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#4:

Solutions:

$$a)\hspace{.2em}\frac{-7}{(x + h - 1)(x - 1)}$$

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#5:

Solutions:

$$a)\hspace{.2em}\frac{1}{\sqrt{x + h - 4}+ \sqrt{x - 4}}$$