Question

evaluate the function ( f(x) = \frac{8}{x} ) at the given values of the independent variable and simplify.

a. ( f(2) )

b. ( f(-8) )

c. ( fleft( \frac{1}{4}
ight) )

a. ( f(2) = ) (simplify your answer.)

b. ( f(-8) = ) (simplify your answer.)

Explanation:

Part a: Evaluate \( f(5) \)

Step 1: Substitute \( x = 5 \) into \( f(x) \)

The function is \( f(x)=\frac{8}{x} \). Substitute \( x = 5 \) into the function: \( f(5)=\frac{8}{5} \)

Step 2: Simplify (already simplified)

The expression \( \frac{8}{5} \) is in its simplest form.

Step 1: Substitute \( x = -4 \) into \( f(x) \)

The function is \( f(x)=\frac{8}{x} \). Substitute \( x = -4 \) into the function: \( f(-4)=\frac{8}{-4} \)

Step 2: Simplify the fraction

Simplify \( \frac{8}{-4} \) by dividing 8 by -4: \( \frac{8}{-4}=-2 \)

Step 1: Substitute \( x = \sqrt{2} \) into \( f(x) \)

The function is \( f(x)=\frac{8}{x} \). Substitute \( x = \sqrt{2} \) into the function: \( f(\sqrt{2})=\frac{8}{\sqrt{2}} \)

Step 2: Rationalize the denominator (optional, but to simplify)

Multiply numerator and denominator by \( \sqrt{2} \) to rationalize: \( \frac{8\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{8\sqrt{2}}{2} \)

Step 3: Simplify the fraction

Simplify \( \frac{8\sqrt{2}}{2} \) by dividing 8 by 2: \( 4\sqrt{2} \)

Answer:

\( \frac{8}{5} \)

Part b: Evaluate \( f(-4) \)