Problem 73 - College Algebra

Question

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2)\). $$ f(x)=3^{x} $$

Step-by-Step Solution

Answer

The values are \(f(-2) = \frac{1}{9}\), \(f(-1) = \frac{1}{3}\), \(f(0) = 1\), \(f(1) = 3\), \(f(2) = 9\).

Step 1: Evaluating f(-2)

To evaluate the function at this point, substitute \(x = -2\) into the function \(f(x) = 3^x\). Thus, we get \(f(-2) = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}\).

Step 2: Evaluating f(-1)

Substitute \(x = -1\) into the function to find \(f(-1)\). This gives us \(f(-1) = 3^{-1} = \frac{1}{3}\).

Step 3: Evaluating f(0)

Substitute \(x = 0\) into the function. Since any non-zero number raised to the power of zero is 1, \(f(0) = 3^0 = 1\).

Step 4: Evaluating f(1)

Substitute \(x = 1\) into the function. We get \(f(1) = 3^1 = 3\).

Step 5: Evaluating f(2)

Finally, substitute \(x = 2\) into the function. Thus, \(f(2) = 3^2 = 9\).

Key Concepts

Function Evaluation

Evaluating a function means finding the output value of a function for a particular input. This is a common step when working with various types of functions, including exponential functions.
In this context, we are given a specific function: \( f(x) = 3^x \). The task is to find what \( f(x) \) equals when specific values are plugged into the equation for \( x \). This is often done to understand how the function behaves across different points.
The process involves substituting the given value of \( x \) into the function and calculating the result. Function evaluation can help verify understanding of the function's pattern, outputs, and growth across different regions.

Powers of Numbers

Powers of numbers, also known as exponents, are a crucial mathematical concept used frequently in exponential functions. In the function \( f(x) = 3^x \), we deal with different powers of the number 3.
The exponent indicates how many times the base (3 in this case) is multiplied by itself. Here's a quick explanation of the powers involved:

• \( 3^2 \): Multiply 3 by itself once, which equals 9.
• \( 3^1 \): The number itself, which is 3.
• \( 3^0 \): Any number to the power of 0 is 1.
• \( 3^{-1} \): Represents a reciprocal, \( \frac{1}{3} \).
• \( 3^{-2} \): The reciprocal of \( 3^2 \), which is \( \frac{1}{9} \).

Understanding these powers is vital because they drastically influence the output of exponential functions, affecting how the graph of the function looks.

Substitution Method

The substitution method is a straightforward technique used to solve problems involving variables, just like our exercise here with the function \( f(x) = 3^x \).
This method involves replacing a variable, often represented as \( x \) in the function, with a given number.
For example:

• For \( f(-2) \): Substitute \( x \) with -2 to calculate \( 3^{-2} \).
• For \( f(0) \): Substitute \( x \) with 0, giving you \( 3^0 = 1 \).
• Continue similarly for other values like 1 and 2.

The substitution method allows us to manually compute results by directly inputting values into the function, which is more straightforward than solving algebraically. This is particularly useful in understanding the value changes in exponential functions across such scenarios.