Question

In Exercises 47-50, use a graphing calculator to graph the function. Then determine whether the function is even, odd, or neither. $$ h(x)=\frac{6}{x^2+1} $$

Short answer

The graph of function \(h(x)=\frac{6}{x^2+1}\) is symmetric about the y-axis and it satisfies the property of even function \(h(-x) = h(x)\). Therefore, the function \(h(x)=\frac{6}{x^2+1}\) is an even function.

Step-by-step solution

Graphing the function h(x)

First, graph the function \(h(x)=\frac{6}{x^2+1}\) using a graphing calculator. From the graph, it can be seen that function is symmetric about y-axis. This suggests that function might be even.

Testing for Even

An even function in mathematics is a function where for every x value in the function, the function of -x is the equal to the function of x. For this function we can substitute -x for x as \(h(-x) = \frac{6}{(-x)^2+1} = \frac{6}{x^2 + 1} = h(x)\). As \(h(x) = h(-x)\), the function is even.

Testing for Odd

To verify if the function isn't uniquely odd too, substitute -x for x and the result should be equal to -1 times the function of x. For this function we can calculate \(h(-x) = \frac{6}{(-x)^2 + 1} = \frac{6} {x^2 + 1} \neq -h(x)\). Therefore, the function is not odd.

Key concepts

Graphing Functions

Visualizing mathematical functions can provide amazing insights and help understand their behavior. When graphing a function like \( h(x) = \frac{6}{x^2 + 1} \), a graphing calculator becomes a helpful tool. A graphing calculator presents the function's graph so we can visually see patterns and symmetries.

To graph a function, you input the equation into the calculator, set a reasonable range for the x- and y-axes, and then view the graph. In this case, graphing \( h(x) = \frac{6}{x^2 + 1} \) reveals that the curve is symmetric about the y-axis. Paying close attention to these visual details helps identify characteristics like symmetry, maximum and minimum points, and behavior at large or small values of x.

Function Symmetry

Function symmetry is an essential aspect to consider when studying functions. It explains how a function's graph behaves regarding reflections, rotations, or translations. There are two main symmetries to consider: even and odd symmetry.

• Even functions are symmetric about the y-axis. This means that for every point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\). Mathematically, this means \( f(x) = f(-x) \).
• Odd functions have rotational symmetry about the origin. If a function is odd, for every point \((x, y)\), the point \((-x, -y)\) also lies on the graph. This yields the condition \( f(x) = -f(-x) \).

By examining a graph, these symmetries can be used to determine quickly whether a function is even, odd, or neither.

Graphing Calculator

A graphing calculator is an essential tool in handling complex functions and understanding their graphs. It allows for efficient calculations and provides a clear view of the function’s behavior.

To use a graphing calculator:

• Enter the function equation.
• Select an appropriate window for your x and y values.
• Observe the graph to identify features like intercepts, asymptotes, and symmetry.

Using \( h(x) = \frac{6}{x^2 + 1}\), you can inspect the symmetry of the graph. This particular function clearly shows symmetry about the y-axis, confirming it as an even function.

Polynomial Functions

While \( h(x) = \frac{6}{x^2 + 1} \) isn’t a polynomial function, understanding polynomials helps in recognizing their behavior and symmetry.

• Polynomials are expressions that involve sums of powers of a variable, like \( ax^n + bx^{n-1} + \cdots + c \).
• Determining evenness or oddness in polynomials is straightforward. If all terms of a polynomial have even powers, it is an even function.
• If all terms have odd powers, then it’s an odd function.

Learning how these rules apply makes it easier to explore more complex functions. Polynomials are the stepping stones to understanding more intricate algebraic functions.