Question

If \(f(x)=a^{x},\) show that \(f(-x)=\frac{1}{f(x)}\).

Short answer

Since \( a^{-x} = \frac{1}{a^x} \), and \( a^x = f(x) \), we have \( f(-x) = \frac{1}{f(x)} \).

Step-by-step solution

Understanding the Given Function

The function given is an exponential function defined as \( f(x) = a^x \), where \( a \) is a constant base and \( x \) is the exponent.

Express the Negative Argument

To find \( f(-x) \), replace \( x \) with \( -x \) in the function. Therefore, \( f(-x) = a^{-x} \).

Using the Negative Exponent Rule

Recall the negative exponent rule, which states \( a^{-x} = \frac{1}{a^x} \). Apply this rule to the expression \( a^{-x} \).

Substitute Back to the Function

Since \( a^{-x} = \frac{1}{a^x} \), we can substitute back to get \( f(-x) = \frac{1}{a^x} \).

Relate to the Original Function

Recognize that \( a^x \) is just \( f(x) \). Therefore, \( f(-x) = \frac{1}{f(x)} \).

Key concepts

Negative Exponents

Negative exponents are special types of exponents that essentially invert the base of the term they are applied to. If you see a term with a negative exponent, like \( a^{-x} \), it can be rewritten using the rule for negative exponents: \( a^{-x} = \frac{1}{a^x} \). This is a crucial point to understand for exponential functions.
For example, \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \). Notice how the base stays the same but moves to the denominator. Understanding this property helps us manipulate and simplify exponential expressions effectively.

Function Evaluation

Function evaluation means substituting a specific value for the variable in the function. For example, if you have a function \( f(x) = a^x \), and you want to find \( f(2) \), you substitute 2 for \( x \) to get \( a^2 \).
In our exercise, the function is \( f(x) = a^x \), and we need to evaluate it at \( -x \). By replacing \( x \) with \( -x \), we get \( f(-x) = a^{-x} \). This substitution is a straightforward way to see how the function behaves at a negative input value.
Breaking down the function evaluation process step-by-step makes it easier to manage complex functions and understand their behavior under different conditions.

Properties of Exponents

Properties of exponents are essential rules that help simplify and manipulate expressions involving exponents. Some key properties include:

• **Product of Powers Property**: \( a^m \times a^n = a^{m+n} \)
• **Quotient of Powers Property**: \( \frac{a^m}{a^n} = a^{m-n} \)
• **Power of a Power Property**: \( (a^m)^n = a^{mn} \)
• **Negative Exponent Property**: \( a^{-n} = \frac{1}{a^n} \)

These properties are very useful for simplifying exponential expressions. In our exercise, we used the negative exponent property to show that \( f(-x) = \frac{1}{f(x)} \). Recognizing and applying these properties will help you solve problems more efficiently.