Question

evaluate the function at the given values of the independent variable and simplify.
$f(x) = \frac{7x^2 - 1}{x^2}$
(a) $f(3)$ (b) $f(-3)$ (c) $f(-x)$
(a) $f(3)=\square$ (type an integer or a fraction. simplify your answer )

Explanation:

Step1: Substitute $x=3$ into $f(x)$

$f(3)=\frac{7(3)^2 - 1}{(3)^2}$

Step2: Calculate squares in numerator/denominator

$f(3)=\frac{7(9) - 1}{9}$

Step3: Compute numerator terms

$f(3)=\frac{63 - 1}{9}$

Step4: Simplify the fraction

$f(3)=\frac{62}{9}$

Step1: Substitute $x=-3$ into $f(x)$

$f(-3)=\frac{7(-3)^2 - 1}{(-3)^2}$

Step2: Calculate squares (negative squared is positive)

$f(-3)=\frac{7(9) - 1}{9}$

Step3: Compute numerator terms

$f(-3)=\frac{63 - 1}{9}$

Step4: Simplify the fraction

$f(-3)=\frac{62}{9}$

Step1: Substitute $x=-x$ into $f(x)$

$f(-x)=\frac{7(-x)^2 - 1}{(-x)^2}$

Step2: Simplify squared negative terms

$f(-x)=\frac{7x^2 - 1}{x^2}$

Step3: Split the fraction (optional simplification)

$f(-x)=7 - \frac{1}{x^2}$

Answer:

(a) $\frac{62}{9}$
(b) $\frac{62}{9}$
(c) $\frac{7x^2 - 1}{x^2}$ or $7 - \frac{1}{x^2}$