Question

Using words and figures, explain why the range of \(f(x)=x^{n},\) where \(n\) is a positive odd integer, is all real numbers. Explain why the range of \(g(x)=x^{n},\) where \(n\) is a positive even integer, is all nonnegative real numbers.

Short answer

Question: Explain why the range of the function \(f(x) = x^n\) is all real numbers when the exponent \(n\) is a positive odd integer, and why the range of the function \(g(x) = x^n\) is all nonnegative real numbers when the exponent \(n\) is a positive even integer. Answer: The range of the function \(f(x) = x^n\) is all real numbers when \(n\) is a positive odd integer because it is an odd function, meaning it is symmetric with respect to the origin, and the function will produce positive values for negative \(x\) and negative values for positive \(x\). In contrast, the range of the function \(g(x) = x^n\) is all nonnegative real numbers when \(n\) is a positive even integer because it is an even function, meaning it is symmetric with respect to the y-axis and will always produce nonnegative values.

Step-by-step solution

Understand the Difference Between Odd and Even Functions

In mathematical terms, a function is odd if \(f(-x) = -f(x)\) and even if \(f(-x) = f(x)\). In other words, an odd function is symmetric with respect to the origin, while an even function is symmetric with respect to the y-axis. This is an important distinction, as it will help explain the difference in ranges for these two types of functions.

Consider Odd Function

For an odd function, if \(f(x)=x^n\) where \(n\) is a positive odd integer, we have \(f(-x)=(-x)^n=-x^n\). This means that when the exponent is odd, the value of the function at \(x\) is always the negative of its value at \(-x\). Therefore, when \(x\) is negative, \(f(x)\) is positive, and when \(x\) is positive, \(f(x)\) is negative, and \(f(x)\) becomes zero at \(x=0\). This tells us that the function \(f(x) = x^n\) will cover all real numbers when \(n\) is an odd integer.

Consider Even Function

Now consider the case of an even function, where \(g(x) = x^n\) and \(n\) is a positive even integer. In this case, we have \(g(-x)=(-x)^n=x^n\). This means that when the exponent is even, the value of the function at \(x\) is the same as its value at \(-x\). This tells us that the function \(g(x) = x^n\) will cover only nonnegative real numbers when \(n\) is an even integer, since no matter if \(x\) is positive or negative, the function will always give us a nonnegative result. Note that \(g(x)\) becomes zero at \(x=0\).

Conclusion

In summary, for the function \(f(x) = x^n\), if \(n\) is a positive odd integer, its range is all real numbers because it is symmetric with respect to the origin, and the function will produce positive values for negative \(x\) and negative values for positive \(x\). On the other hand, for the function \(g(x) = x^n\), if \(n\) is a positive even integer, its range is all nonnegative real numbers because it is symmetric with respect to the y-axis and will always produce nonnegative values.

Key concepts

Odd Functions

Odd functions have a special symmetry in mathematics. A function is considered odd if it satisfies the equation \(f(-x) = -f(x)\). This type of symmetry means that the graph of the function is a mirror image in both the positive and negative quadrants of the coordinate plane when looked around the origin (0,0).

For example, if you have \(f(x) = x^n\) where \(n\) is an odd positive integer like 1, 3, or 5, the function's output will behave in a specific way:

• For negative values of \(x\), \(f(x)\) will be negative if the input \(x\) is negative.
• For positive values of \(x\), \(f(x)\) will be positive if \(x\) is positive.
• At \(x = 0\), the function \(f(x)\) equals zero: \(f(0) = 0^n = 0\).

Because of this symmetry about the origin, odd functions such as \(f(x) = x^3\) will have a range of all real numbers from negative infinity to positive infinity. This means they can output any value depending on the input.

Even Functions

An even function is quite distinct from an odd function. For a function to be even, it must satisfy the condition \(f(-x) = f(x)\). This signifies that the graph of the function has symmetry around the y-axis. This makes even functions quite predictable, as their outputs remain the same for both positive and negative inputs of an equal absolute value.

Take for instance the function \(g(x) = x^n\) where \(n\) is an even positive integer such as 2, 4, or 6:

• For any positive or negative value of \(x\), the function always produces a non-negative output.
• The value of \(g(x)\) is the same as \(g(-x)\), hence the symmetry.
• The point \(x = 0\) will result in \(g(0) = 0^n = 0\), keeping the value still nonnegative.

As a result of this symmetry about the y-axis, even functions such as \(g(x) = x^2\) have a range restricted to nonnegative real numbers. This means they can only give values ranging from zero to positive infinity.

Range of Functions

Understanding the range of a function is crucial for grasping the possible output values based on all potential inputs. When discussing the function \(f(x) = x^n\), knowing whether \(n\) is even or odd helps in determining the range effectively.

- For odd integers \(n\): The function \(f(x) = x^n\) results in a range covering all real numbers. This is due to the origin symmetry, which leads to both negative and positive outputs for appropriate inputs.- For even integers \(n\): The function \(g(x) = x^n\) constrains its range to nonnegative real numbers. Basically, whether you input a positive or a negative \(x\), the output will be nonnegative due to the y-axis symmetry.

The range thus informs us of the breadth of values a function can output, guiding mathematicians and students alike in analyzing and predicting function behavior. Understanding how symmetry affects the range of a function provides a foundational skill in addressing many mathematical problems.