### Home > CALC > Chapter 2 > Lesson 2.3.1 > Problem2-108

2-108.

Given: $h ( x ) = | x ^ { 2 } - 1 |$

1. Rewrite $h(x)$ as a piecewise function.

Notice how this $y = |x^² − 1|$ graph differs from $y = x^² − 1$. From this we can determine that there will be three pieces, and we already know the first and last piece.

The middle piece is a vertical reflection of $y = x^² − 1$.

$x^² − 1, x < −1$
$y = −x^² + 1, −1 ≤ x ≤ 1$
$x^² − 1, x > 1$

2. Using set notation, state the domain and range of $h(x)$.

Consider how the range of $h(x)$ differs from the range of $y = x^² − 1$.

3. Estimate $A(h, −1 ≤ x ≤ 3)$ by any method.

A trapezoidal sum is an option.

4. Write a Riemann sum to approximate $A(h, −1 ≤ x ≤ 3)$ with $24$ rectangles of equal width and evaluate the sum.

The height is determined by the function.

Use the eTool below to help answer each part:
Click on the link to the right to view the full version of the eTool. Calc 2-108 HW eTool