Problem 49 - Algebra A Combined Function

Question

If \(f(x)=3^{x}\), find each value. give an exact answer and a two-decimal-place approximation. See Sections 8.2 and \(10.2 .\) $$ f(2) $$

Step-by-Step Solution

Answer

The exact answer is 9, and the approximate value is 9.00.

Step 1: Substitute in given function

To solve for \( f(2) \), substitute \( x = 2 \) into the function \( f(x) = 3^x \). This gives us \( f(2) = 3^2 \).

Step 2: Evaluate the power

Calculate \( 3^2 \). Raise 3 to the power of 2, which results in \( 9 \).

Step 3: Exact answer

The exact value of \( f(2) \) is \( 9 \) because \( 3^2 = 9 \).

Step 4: Decimal approximation

Since the result of \( 3^2 \) is already an integer, the two-decimal-place approximation of 9 is simply \( 9.00 \).

Key Concepts

Substitution Method

The substitution method is a mathematical technique where you replace a variable with a given value.
This simplifies the expression and allows you to solve or evaluate the function easily.
In our original exercise, we are given the function \( f(x) = 3^x \) and need to find \( f(2) \). To do so, you simply substitute \( x = 2 \) into the function.
This changes the expression \( f(x) = 3^x \) to \( f(2) = 3^2 \).
Substitution is essential for pinpointing specific values of a function quickly.

• Identify the variable value.
• Replace the variable with its given number.
• Simplify the expression to find the answer.

Practicing substitution helps to deepen understanding of functions and how variables interact within them.

Evaluating Powers

Evaluating powers is about calculating the result of an exponential expression.
In this context, a power consists of a base (like 3) and an exponent (like 2).In our example, after substituting \( x = 2 \) into \( f(x) = 3^x \), we are left with \( 3^2 \). To evaluate this power, remember these points:

• The base is the number you multiply by itself.
• The exponent denotes how many times you multiply the base.

For \( 3^2 \), multiply 3 by itself to get:\[3 \times 3 = 9\]This step is fundamental when working with exponential functions, providing the exact values we need for further calculations.

Exact Answer

An exact answer is a precise value without any rounding or approximation.After calculating \( 3^2 \), we derive an exact answer of 9.
Since the operation involves whole numbers with clear results, the exact value remains a whole number without fractions or decimals. Exact answers are crucial in assessing the correctness of calculations.
They serve as a benchmark before any approximations are made.

Decimal Approximation

Decimal approximation involves rounding a number to a specified decimal place.
It provides a simplified version without significantly altering the original value.In the case of \( f(2) = 9 \), the exact answer is a whole number.
Therefore, its decimal approximation is straightforward and results in \( 9.00 \).

• Locate the decimal place to which you need to round.
• If no decimals exist, add zeros to reach the necessary decimal places.

Decimal approximations are valuable in scenarios where precision to a certain decimal is needed, facilitating clear communication and comparison with other numbers.