Question

evaluate the function at the given values of the independent variable and simplify
$f(x)=\frac{6x^2 - 1}{x^2}$
(a) $f(2)$ (b) $f(-2)$ (c) $f(-x)$
(a) $f(2)=\frac{23}{4}$ (type an integer or a fraction. simplify your answer.)
(b) $f(-2)=\frac{23}{4}$ (type an integer or a fraction. simplify your answer.)
(c) $f(-x)=\square$ (simplify your answer.)

Explanation:

Step1: Substitute -x into the function

We have the function \( f(x)=\frac{6x^{2}-1}{x^{2}} \). To find \( f(-x) \), we replace every \( x \) in the function with \( -x \). So we get \( f(-x)=\frac{6(-x)^{2}-1}{(-x)^{2}} \).

Step2: Simplify the exponents

Recall that \( (-a)^{2}=a^{2} \) for any real number \( a \). So \( (-x)^{2}=x^{2} \). Substituting this into the expression for \( f(-x) \), we have \( f(-x)=\frac{6x^{2}-1}{x^{2}} \). We can also rewrite this as \( 6 - \frac{1}{x^{2}} \) by splitting the fraction: \( \frac{6x^{2}}{x^{2}}-\frac{1}{x^{2}} = 6-\frac{1}{x^{2}} \), but the form \( \frac{6x^{2}-1}{x^{2}} \) is also simplified.

Answer:

\( \frac{6x^{2}-1}{x^{2}} \) (or \( 6 - \frac{1}{x^{2}} \))