Problem 13 - 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. $$f(3)$$

Step-by-Step Solution

Answer

The value of \( f(3) \) is 27.

Step 1: Identify the Function

The function given is \( f(x) = 3^x \). We need to find the value when \( x = 3 \).

Step 2: Substitute the Value

Substitute \( x = 3 \) into the function: \( f(3) = 3^3 \).

Step 3: Calculate the Exponentiation

Compute \( 3^3 \) which equals \( 3 \times 3 \times 3 \).

Step 4: Solve the Multiplication

First, calculate \( 3 \times 3 = 9 \), then \( 9 \times 3 = 27 \).

Step 5: Conclusion

Thus, the value of \( f(3) \) is 27.

Key Concepts

Function Evaluation

Function evaluation is a key concept when working with mathematical functions. It involves determining the value of a function when a specific input, or variable, is used. For example, if you are given a function \( f(x) = 3^x \), and you want to find \( f(3) \), you are performing function evaluation. Here, the input is \( x = 3 \), and you plug this value into the function to get the result.

A few simple steps can help you complete a function evaluation:

• Identify the function and the specific input value you need to evaluate.
• Substitute the input value into the function wherever the variable appears.
• Simplify the expression, following the correct order of operations.

This allows you to find the function's output for the given input, which is an essential component of understanding how different functions behave.

Exponentiation

Exponentiation is a mathematical operation form that involves two numbers, the base and the exponent. In an expression like \( 3^3 \), the number 3 is the base, and it is raised to the power of the exponent, which is also 3. This means you multiply the base by itself as many times as indicated by the exponent.

To perform the exponentiation operation correctly:

• Start with the base, e.g., 3.
• Multiply the base by itself for the number of times indicated by the exponent. For \( 3^3 \), you multiply 3 three times: \( 3 \times 3 \times 3 \).
• Solve the multiplication sequentially to get the result, so \( 3^3 = 27 \).

Understanding exponentiation allows us to work with powers and roots, which are important in many areas of mathematics, including algebra and calculus.

Substitution in Functions

Substitution is a technique used to evaluate functions by replacing the variable in a function with a specific number. This is vital in determining particular values of functions. In our exercise, substituting \( x = 3 \) into the function \( f(x) = 3^x \) transformed it into \( 3^3 \).

The steps for substitution in functions include:

• Identify which variable in the function will be replaced.
• Replace the variable with the given number or expression, carefully placing it in the correct parts of the function.
• Compute the rest of the expression using the substituted values to simplify and find the solution.

Mastering substitution in functions is crucial for tackling complex equations and applied problems in mathematics. It helps bridge abstract formulas into concrete and usable results.