Problem 50 - 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 \(10.2 .\) $$ f(0) $$

Step-by-Step Solution

Answer

The exact value of \( f(0) \) is 1, and its two-decimal-place approximation is 1.00.

Step 1: Understand the Function

The function given is \( f(x) = 3^x \). This is an exponential function where 3 is the base, and \( x \) is the exponent. Our task is to evaluate this function at \( x = 0 \).

Step 2: Substitute the Given Value

Replace \( x \) in the function \( f(x) \) by 0. So, we evaluate \( f(0) = 3^0 \).

Step 3: Evaluate the Exponential Expression

Any non-zero number raised to the power of 0 is 1. Therefore, \( 3^0 = 1 \).

Step 4: Provide the Exact Value

The exact value of \( f(0) \) is 1. Since 1 is already a simple whole number, it doesn't require further simplification.

Step 5: Provide the Two-Decimal-Place Approximation

The value 1 is exact and also an integer, so its approximation to two decimal places is 1.00.

Key Concepts

Substitution Method

The substitution method is a key technique when dealing with functions and equations. It involves replacing a variable with a specific value to simplify the problem or to evaluate the function at that specific point. In the context of exponential functions like \[ f(x) = 3^x \], substitution allows us to determine the value of the function for a given number.

Here's how it works:

• Start by identifying the variable in the function. In this case, it's \( x \).
• Determine the specific value you want to substitute in place of \( x \). For example, to find \( f(0) \), substitute 0 for \( x \).
• After substitution, evaluate the expression to find the result. So, for our example, it's \( 3^0 \).

This method is straightforward but powerful, especially when applied to functions with varying degrees of complexity.

Zero Exponent Rule

The zero exponent rule is a fundamental aspect of exponents that simplifies your calculations when the exponent is zero. It states that any non-zero number raised to the zero power is equal to 1.

This principle applies broadly across all non-zero numbers.

• For example, \( 3^0 = 1 \), just as \( 5^0 = 1 \), \( (-2)^0 = 1 \), and so on.

Understanding and applying the zero exponent rule is essential when evaluating functions, particularly exponential functions. It's particularly useful in simplifying expressions where at least one part of the expression contains a zero exponent.This rule not only helps in simplifying your work but also ensures that you fully grasp the nature of exponents. Remember, the key aspect here is that the base must not be zero for this rule to be applicable.

Exact Value Calculation

Exact value calculation focuses on providing precise results rather than approximations. It is particularly vital in mathematics when dealing with functions, such as exponential functions, where the powers are precisely defined.

In our exercise, after using the substitution method and applying the zero exponent rule, we arrive at the exact value of \( 3^0 \), which is 1. This value is absolute and doesn't need further approximation.

• Exact values are essential in both theoretical and applied mathematics for accurate results.
• They preserve the integrity of mathematical expressions and often serve as reference points when further calculations or approximations are required.

For instance, when an exact value like 1 is asked for any approximate measurement to two decimal places, you still maintain the precision: 1.00. This operation doesn't alter the value but aligns with representation standards for clarity and consistency.