Let 𝑓(𝑥) = 5𝑥2−1
𝑥2 . Evaluate the function for the given values/expressions and simplify.
a) f(3) = 
b) f(0) = 
c) f(-x) = 
d) f(1+x) = 

We are given the function:

\[
f(x) = \frac{5x^2 - 1}{x^2}
\]

We will evaluate \( f(x) \) for each of the specified values.

---

### a) \( f(3) \)

Substitute \( x = 3 \) into the function:

\[
f(3) = \frac{5(3)^2 - 1}{(3)^2} = \frac{5(9) - 1}{9} = \frac{45 - 1}{9} = \frac{44}{9}
\]

So, \( f(3) = \frac{44}{9} \).

---

### b) \( f(0) \)

Substitute \( x = 0 \) into the function:

\[
f(0) = \frac{5(0)^2 - 1}{(0)^2}
\]

However, division by 0 is undefined. Thus, \( f(0) \) is undefined.

---

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

Substitute \( x = -x \) into the function:

\[
f(-x) = \frac{5(-x)^2 - 1}{(-x)^2}
\]

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

\[
f(-x) = \frac{5x^2 - 1}{x^2}
\]

So, \( f(-x) = f(x) = \frac{5x^2 - 1}{x^2} \).

---

### d) \( f(1+x) \)

Substitute \( x = 1 + x \) into the function:

\[
f(1+x) = \frac{5(1 + x)^2 - 1}{(1 + x)^2}
\]

First, expand \( (1 + x)^2 \):

\[
(1 + x)^2 = 1 + 2x + x^2
\]

Now substitute this into the function:

\[
f(1+x) = \frac{5(1 + 2x + x^2) - 1}{1 + 2x + x^2}
\]

Simplify the numerator:

\[
5(1 + 2x + x^2) - 1 = 5 + 10x + 5x^2 - 1 = 4 + 10x + 5x^2
\]

Thus:

\[
f(1+x) = \frac{4 + 10x + 5x^2}{1 + 2x + x^2}
\]

---

Let me know if you need more details or clarifications!

### Expanding Understanding:
1. What happens if \( f(x) \) is evaluated at any other negative number?
2. How would the domain of \( f(x) \) change if it were defined as \( \frac{5x^2 - 1}{x} \)?
3. What is the symmetry of \( f(x) \)? Is it even, odd, or neither?
4. How does the graph of \( f(x) \) behave as \( x \to \infty \)?
5. How would you find the derivative of \( f(x) \)?

### Tip:
When simplifying fractions, always check for common factors in the numerator and denominator to reduce the expression fully.

Let f(x) = \frac{5x^2 - 1}{x^2}. Evaluate the function for the given values/expressions and simplify. a) f(3) = b) f(0) = c) f(-x) = d) f(1+x) =

Learn how to evaluate and simplify the rational function f(x) = (5x^2 - 1)/x^2 for different values of x. This step-by-step guide covers key concepts like rational functions, even and odd function properties, and undefined expressions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation