Question

Determine algebraically whether each function is even, odd, or neither. \(g(x)=10-x^{2}\)

Short answer

The function \( g(x) = 10 - x^2 \) is even, because \( g(-x) = g(x) \).

Step-by-step solution

Understand the Definitions

A function is even if for all x in the domain of the function, \( f(-x) = f(x) \). A function is odd if for all x in the domain of the function, \( f(-x) = -f(x) \). If neither condition is satisfied, the function is neither even nor odd.

Substitute \( -x \) into \( g(x) \)

To determine if \( g(x) = 10 - x^2 \) is even or odd, substitute \( -x \) into the function: \ g(-x) = 10 - (-x)^2 \.

Simplify the Expression

Simplify \( g(-x) \: \ g(-x) = 10 - x^2 \). Notice that \( -x^2 = x^2 \). Therefore, \ g(-x) = 10 - x^2 \.

Compare \( g(-x) \) with \( g(x) \)

Compare the simplified expression \( g(-x) \) with the original function \( g(x) \: \ g(-x) = 10 - x^2 = g(x) \).

Determine if the Function is Even, Odd, or Neither

Since \( g(-x) = g(x) \), the function \( g(x) = 10 - x^2 \) is even.

Key concepts

Substitution Method

The substitution method is a straightforward approach used to determine if a function is even, odd, or neither. To start, take the given function and substitute \(-x\) for \(x\). This means you should replace every instance of \(x\) with \(-x\) in the function.
For example, if the original function is \(g(x) = 10 - x^2\), substituting \(-x\) gives us \(g(-x) = 10 - (-x)^2\).
Be meticulous when performing substitutions to avoid errors. This step is crucial for verifying the symmetry properties of the function.

Function Symmetry

Function symmetry helps categorize functions and understand their geometric properties.
An **even function** has y-axis symmetry. This means the graph on one side of the y-axis mirrors the other side. For an even function, \(f(-x) = f(x)\).
Check by comparing the original function and the substituted function. In our problem, \(g(-x) = 10 - x^2 = g(x)\), so \(g(x)\) is even.
An **odd function** has origin symmetry, meaning rotating the graph 180 degrees around the origin leaves it unchanged. For an odd function, \(f(-x) = -f(x)\).
If neither condition is met, the function is neither even nor odd.

Function Properties

Understanding the properties of a function allows us to predict its behavior without graphing it.
In algebra, recognizing even and odd functions can simplify solving equations and integrating functions over symmetric intervals.
An even function, like \(g(x) = 10 - x^2\), shows symmetry about the y-axis. It means that calculating the function's value at \(-x\) is the same as at \(x\).
These properties also help in calculus, where integrating an even function over a symmetric interval simply doubles the integral from 0 to the endpoint, given that the function is symmetric and defined for the entire interval.