Problem 16 - Precalculus

Question

For the functions \(f(x)=3^{x}, g(x)=\left(\frac{1}{16}\right)^{x},\) and \(h(x)=10^{x+1},\) find the function value at the indicated points. $$f(-2)$$

Step-by-Step Solution

Answer

The function value \(f(-2)\) is \(\frac{1}{9}\).

Step 1: Understanding the Function

We are given the function \(f(x) = 3^x\). To find \(f(-2)\), we will substitute \(-2\) into \(f(x)\).

Step 2: Substitute the Given Value

Substitute \(-2\) into the function \(f(x) = 3^x\) to find \(f(-2)\). This gives us:\[ f(-2) = 3^{-2} \]

Step 3: Evaluate the Exponent

Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent. Thus:\[ 3^{-2} = \frac{1}{3^2} \]

Step 4: Calculate the Power

Now, compute \(3^2\):\[ 3^2 = 9 \]

Step 5: Simplify the Expression

Using the result from Step 4, substitute back into the reciprocal:\[ \frac{1}{3^2} = \frac{1}{9} \]

Step 6: Write the Final Answer

Thus, the function value \(f(-2)\) is:\[ f(-2) = \frac{1}{9} \]

Key Concepts

Negative Exponents

When dealing with exponential functions, you might encounter negative exponents. A negative exponent signifies that you should take the reciprocal of the base and then raise it to the positive version of the exponent. For example, if you have a negative exponent like in the expression \[3^{-2}\], it instructs you to think of it as the reciprocal of \(3\) raised to the \(2^{nd}\) power. This transforms the expression into \(\frac{1}{3^2}\).

Negative exponents are helpful in simplifying complex expressions, especially when they are part of division. Remember:

• Negative exponent: reciprocal of the base raised to a positive exponent.
• Turns division problems into multiplication by the reciprocal.
• Provides an easy way to manage expressions that involve powers of values smaller than one.

Reciprocal

The reciprocal is an important concept in mathematics when dealing with fractions and exponents. A reciprocal of a number simply means flipping its numerator and denominator. Thus, the reciprocal of a number \(a\) is \(\frac{1}{a}\). In the context of exponential functions, especially with negative exponents, the idea of reciprocals plays a crucial role.

When you have a negative exponent, the reciprocal essentially becomes part of the calculation process. For instance, for \(3^{-2}\):

• First, interpret the negative exponent (as explained above).
• Express it as a reciprocal \(\frac{1}{3^2}\).
• Then compute the value of \(3^2\), which is \(9\).
• Finally, the reciprocal calculation yields \(\frac{1}{9}\).

Understanding reciprocals not only simplifies calculations but also provides clarity when simplifying expressions with fractions.

Function Evaluation

Function evaluation is the process of substituting a specific input into a function to compute the corresponding output. In exponential functions, this involves replacing the variable \(x\) with a given number and performing the necessary arithmetic operations to find the result. Here's how to evaluate an exponential function step-by-step:

1. **Identify the function and the variable.** Ensure you know what function you're working with. For instance, \(f(x) = 3^x\).
2. **Substitute the variable with a given value.** Say you're asked to find \(f(-2)\), replace \(x\) with \(-2\), giving you \(3^{-2}\).
3. **Solve the expression.** Use the properties of exponents (like converting a negative exponent to a reciprocal) to simplify and get the result, \(\frac{1}{9}\).
4. **Interpret the result.** The value derived from this process, \(\frac{1}{9}\) in this case, is the output of the function at the specific input, \(-2\).

By following these steps, you can evaluate any function quickly and accurately. Function evaluation is a vital skill in math that helps you to understand how a function behaves for different inputs.