Problem 13 - Elementary and Intermediate Algebra

Question

Complete the table of function values shown here. $$ f(x)=5^{x} $$ (TABLE CANNOT COPY)

Step-by-Step Solution

Answer

Calculate \( f(x) \) for each \( x \) using \( f(x) = 5^x \) and fill in the table.

Step 1: Understand the given function

The function given is \( f(x) = 5^x \). This means for any value of \( x \), \( f(x) \) is simply \( 5 \) raised to the power of \( x \).

Step 2: Calculate \( f(x) \) for each \( x \) value in the table

You need to evaluate \( f(x) = 5^x \) for the various \( x \) values provided. For example, if the table includes \( x = 0, 1, 2, \) etc., calculate by substituting \( x \) into the function.

Step 3: Calculate \( f(0) \)

Since any non-zero number raised to the power of 0 is 1, \( f(0) = 5^0 = 1 \).

Step 4: Calculate \( f(1) \)

Using the function, \( f(1) = 5^1 = 5 \).

Step 5: Calculate \( f(2) \)

Using the function, \( f(2) = 5^2 = 25 \).

Step 6: Calculate \( f(3) \)

Using the function, \( f(3) = 5^3 = 125 \).

Step 7: Compile the results into the table

Fill in the table with your computed \( f(x) \) values: \( x = 0 \), \( f(x) = 1 \); \( x = 1 \), \( f(x) = 5 \); \( x = 2 \), \( f(x) = 25 \); \( x = 3 \), \( f(x) = 125 \).

Key Concepts

Function Evaluation

Function evaluation is the process of calculating the output of a function for a specific input. It's like a chef using a recipe. Just input the specified ingredients in the method, and you'll get a result. In mathematical terms, we substitute a number for the variable and then perform the calculations to find the function's value. For our function, \( f(x) = 5^x \), the role of the ingredient is played by the variable \( x \).
To evaluate, substitute the chosen number for \( x \), and solve the equation. For instance:

• When \( x = 0 \), evaluate \( f(0) \) by calculating \( 5^0 \), which equals 1.
• When \( x = 1 \), evaluate \( f(1) \) by calculating \( 5^1 \), which equals 5.

Evaluating functions allows us to understand the behavior of the function across different inputs, and it's a fundamental skill in understanding more complex mathematical concepts.

Powers of a Number

Understanding powers of a number is crucial when dealing with exponential functions. A power refers to the number of times you multiply a number by itself. In our function \( f(x) = 5^x \), 5 is the base, and \( x \) is the exponent.
Here are important points to remember about powers:

• \( 5^0 = 1 \). Any non-zero number raised to the power of zero equals 1. It's like resetting back to the start.
• \( 5^1 = 5 \). Any number raised to the power of one equals the number itself.
• \( 5^2 = 25 \). This means multiplying 5 by itself once: \( 5 \times 5 \).
• \( 5^3 = 125 \). This means multiplying 5 by itself twice: \( 5 \times 5 \times 5 \).

Exponential growth is displayed prominently in functions like \( f(x) = 5^x \), demonstrating how quickly values can increase as the exponent grows.

Function Table

A function table helps us visualize and compile the results of function evaluations for different values of the variable. Think of it as a summary sheet that shows how different \( x \) values affect the function output.
For the given function \( f(x) = 5^x \), a function table might contain these values:

• When \( x = 0 \), \( f(x) = 1 \)
• When \( x = 1 \), \( f(x) = 5 \)
• When \( x = 2 \), \( f(x) = 25 \)
• When \( x = 3 \), \( f(x) = 125 \)

Such a table supports better understanding by allowing us to see patterns in how the function behaves. It is especially useful when navigating complex calculations, providing an at-a-glance view of important function results.