Question

Evaluate the function at the given values of the independent variable and simplify.
f left parenthesis x right parenthesis equals StartFraction 3 x squared minus 1 Over x squared EndFraction
​(a) f left parenthesis 4 right parenthesis    ​(b) f left parenthesis negative 4 right parenthesis    ​(c) f left parenthesis negative x right parenthesis
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Part 1
​(a) f left parenthesis 4 right parenthesisequals
  
enter your response here ​(Type an integer or a fraction. Simplify your​ answer.)

Answer 1

To evaluate the function \( f(x) = \frac{3x^2 - 1}{x^2} \), let's follow through with each part step by step.

### Part (a): \( f(4) \)

First, we substitute \( x = 4 \) into the function:

\[
f(4) = \frac{3(4)^2 - 1}{(4)^2}
\]

Calculating \( (4)^2 = 16 \):

\[
f(4) = \frac{3(16) - 1}{16} = \frac{48 - 1}{16} = \frac{47}{16}
\]

So,

\[
f(4) = \frac{47}{16}
\]

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

Next, we substitute \( x = -4 \) into the function:

\[
f(-4) = \frac{3(-4)^2 - 1}{(-4)^2}
\]

As previously calculated, \( (-4)^2 = 16 \):

\[
f(-4) = \frac{3(16) - 1}{16} = \frac{48 - 1}{16} = \frac{47}{16}
\]

Thus,

\[
f(-4) = \frac{47}{16}
\]

### Part (c): \( f(-x) \)

Now, we evaluate \( f(-x) \):

\[
f(-x) = \frac{3(-x)^2 - 1}{(-x)^2}
\]

Since \( (-x)^2 = x^2 \):

\[
f(-x) = \frac{3(x^2) - 1}{x^2} = \frac{3x^2 - 1}{x^2}
\]

This simplifies to the same original function:

\[
f(-x) = f(x)
\]

### Summary of Results

- (a) \( f(4) = \frac{47}{16} \)
- (b) \( f(-4) = \frac{47}{16} \)
- (c) \( f(-x) = f(x) = \frac{3x^2 - 1}{x^2} \)