Problem 51 - 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 \(10.2 .\) $$ f\left(\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

The value is \( \sqrt{3} \) or approximately 1.73.

Step 1: Identify the Function

The function given is \( f(x) = 3^x \). This is an exponential function where the base is 3, and \( x \) is the exponent.

Step 2: Substitute the Value

We need to find \( f\left(\frac{1}{2}\right) \). Substitute \( x = \frac{1}{2} \) into the function. Thus, it becomes \( f\left(\frac{1}{2}\right) = 3^{\frac{1}{2}} \).

Step 3: Simplify the Expression

The expression \( 3^{\frac{1}{2}} \) represents the square root of 3. Thus, we write \( f\left(\frac{1}{2}\right) = \sqrt{3} \).

Step 4: Calculate Exact Value

The exact value of \( \sqrt{3} \) is \( 3^{1/2} \), commonly expressed as \( \sqrt{3} \).

Step 5: Approximate Decimal Value

To get a decimal approximation, compute \( \sqrt{3} \). Using a calculator, the approximation is \( 1.73 \) rounded to two decimal places.

Key Concepts

Understanding Square Roots

Square roots are a way of expressing a number that, when multiplied by itself, gives the original number. In our exercise, the expression \(3^{\frac{1}{2}}\) is an example of finding the square root because the exponent is \(\frac{1}{2}\). This means we are looking for a number that, when squared, yields 3. This is why \(3^{\frac{1}{2}}\) is the same as \(\sqrt{3}\).
Square roots are intuitive when we think about splitting something into identical halves. For example, the square root of 4 is 2 because \(2 \times 2 = 4\). Conversely, the square root of 3 is not a perfect whole number, hence it is irrational and cannot be expressed exactly as a fraction of two integers. That is why we approximate irrational square roots like \(\sqrt{3}\) to two decimal places for practical use.

Exploring Exponents

Exponents represent repeated multiplication of a base number. They show how many times the base is used as a factor. For instance, \(3^x\) means multiplying 3 by itself \(x\) times.
In our exercise, the exponent was \(\frac{1}{2}\). Fractional exponents have special meanings. In this context, the fraction \(\frac{1}{2}\) means finding a square root. More generally:

• A positive integer exponent, like 3, means multiplying the number by itself so many times, e.g., \(3^3 = 3 \times 3 \times 3 = 27\).
• Negative exponents indicate division, \(3^{-2}\) being equivalent to \(\frac{1}{3^2} = \frac{1}{9}\).
• Fractional exponents, like \(\frac{1}{2}\), indicate roots, with \(3^{\frac{1}{2}}\) being \(\sqrt{3}\).

Understanding these rules helps in simplifying and solving expressions involving exponents.

Decimal Approximation Explained

When numbers involve square roots of non-perfect squares, the results are often irrational. This means the decimal goes on forever without repeating. However, in practical applications, it's helpful to approximate these numbers. For example, in our exercise, \(\sqrt{3}\) is approximated to 1.73.
This decimal approximation is important for both calculator usage and when precision isn't critical. To achieve this, we round to two decimal places. Here's how we simplify this process:

• Using a calculator, compute the square root, which will typically provide many decimal digits.
• Consider the third decimal digit for rounding: If it's 5 or greater, round up the second digit by one. Otherwise, leave it unchanged.
• Thus, \(\sqrt{3} \approx 1.73\) because the subsequent decimals encourage this rounding.

This method offers a balance between precision and practicality in mathematical computations.