https://file.aiquickdraw.com/mathbot/file/1729448376116p565wx3k.jpg

In the image you provided, the function given is \( h(x) = \frac{1}{(x-5)^2 + 4(x-5) + 4} \), and the question asks for the value of \( x \) at which the function \( h(x) \) is undefined.

To solve this, we need to find when the denominator of the function is equal to zero, because division by zero makes the function undefined. The denominator is:

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

The working process shown in the image is as follows:

1. Let \( y = (x - 5) \). Substituting this into the equation, we get:
   \[
   y^2 + 4y + 4 = 0
   \]

2. Solving this quadratic equation, we use factoring or the quadratic formula. The equation factors as:
   \[
   (y + 2)^2 = 0
   \]

3. Solving \( (y + 2)^2 = 0 \) gives \( y = -2 \). Since \( y = x - 5 \), substituting back:
   \[
   x - 5 = -2 \implies x = 3
   \]

So, \( h(x) \) is undefined at \( x = 3 \), which is confirmed in the image.

Would you like further clarification on any steps?

Here are 5 related questions to explore:

1. How do you solve other types of rational functions where the denominator involves higher degree polynomials?
2. What happens if the numerator of a rational function is also zero at the same point where the denominator is zero?
3. How can you use the quadratic formula for non-factorable quadratic equations in rational functions?
4. How can you check the domain of a function graphically?
5. How would you solve rational inequalities involving undefined points?

**Tip:** Always remember, a function is undefined wherever the denominator equals zero, and these points often play a key role in determining the domain of the function.

For what value of x is the function h(x) = 1 / [(x-5)^2 + 4(x-5) + 4] undefined?

Learn how to determine when a rational function is undefined using algebraic techniques. This example shows how to solve a quadratic equation from a rational function's denominator to find undefined points. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation