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Chapter 1: Problem 80

Sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answer algebraically. $$h(x)=x^{2}-4$$

Short Answer

Expert verified
The function \(h(x) = x^{2} - 4\) is an even function.

Step by step solution

01

Sketch the function

First, it's necessary to sketch the graph of the function. This can be done by finding some key points. For example, when \(x = 0\), \(h(x) = -4\), and when \(x = 2\) or \(x = -2\), \(h(x) = 0\). Corresponding to these inputs plot the points on the graph. Then connect these points with a smooth curve. The function \(h(x) = x^{2} - 4\) is a downward parabola.
02

Identify the function

The function is symmetric about the y-axis since it's a parabola. So, according to the definition, it might be an even function since all the x values are squared, causing them to be positive. But we can't conclude this just from the shape of the graph.
03

Verify algebraically

According to the definition of even and odd function, substitute \(x\) with \(-x\) in our function and simplify:\(h(-x) = (-x)^{2}-4 = x^{2} - 4 = h(x)\). Hence the function remains the same, thus verifying our previous prediction that it's an even function.

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