Problem 52 - 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

Exact: \(\sqrt{3}\); Approximation: \(1.73\).

Step 1: Substitute the Input Value

To find \(f\left(\frac{1}{2}\right)\), substitute \(x = \frac{1}{2}\) into the function: \[f(x) = 3^{x} \Rightarrow f\left(\frac{1}{2}\right) = 3^{\frac{1}{2}}.\]

Step 2: Express as a Square Root

The expression \(3^{\frac{1}{2}}\) can be rewritten using the square root notation: \[3^{\frac{1}{2}} = \sqrt{3}.\]

Step 3: Calculate the Square Root

Using a calculator, approximate \(\sqrt{3}\) to two decimal places:\[\sqrt{3} \approx 1.73.\]

Step 4: Present the Exact and Approximate Values

The exact value of \(f\left(\frac{1}{2}\right)\) is \(\sqrt{3}\). The approximate value, rounded to two decimal places, is \(1.73\).

Key Concepts

Square Roots

When dealing with exponential functions like \( f(x) = 3^x \), understanding square roots becomes essential, especially when evaluating expressions like \( 3^{\frac{1}{2}} \). A square root is essentially a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \).

In mathematical terms, the square root of a number \( a \) is denoted as \( \sqrt{a} \). In our exercise, when we substitute \( x = \frac{1}{2} \) into the exponential function, we get \( 3^{\frac{1}{2}} \), which is equivalent to \( \sqrt{3} \). This is an exact expression of the value.

• It simplifies the understanding of the exponent \( \frac{1}{2} \) as a square root operation.
• Using square roots helps in representing non-whole number exponents in a more intuitive way.

Once we understand square roots, we see that \( 3^{\frac{1}{2}} \) is merely asking us to consider the square root of 3, which can further be approximated for practical use.

Approximation Techniques

Approximating values is a key skill when the exact answer involves irrational numbers, like square roots. Since square roots often result in long, non-repeating decimals, calculators are used to find a useful approximation.

In our example, the exact value of \( \sqrt{3} \) is calculated to be approximately \( 1.7320508...\) in decimal form. For practical purposes, especially when precision isn't crucial, we often round this to two decimal places: \( 1.73 \).

• Rounding involves reducing the number of decimal places while maintaining closeness to the actual number.
• This is especially useful in fields like science and engineering where exact values can oversimplify the models.

Approximation is an invaluable tool because it simplifies complex calculations, allowing for easier interpretation and real-world application.

Function Evaluation

Function evaluation involves substituting a given input value into a function to find the corresponding output. For example, in the function \( f(x) = 3^x \), we want to evaluate it at \( x = \frac{1}{2} \).

To do this, follow these steps:

• Identify the given function, which is \( f(x) = 3^x \).
• Substitute \( x = \frac{1}{2} \) into the function, resulting in \( f\left(\frac{1}{2}\right) = 3^{\frac{1}{2}} \).
• Recognize this expression represents a square root, \( \sqrt{3} \).

Once this substitution is completed, you've evaluated the function at the given point. The result of function evaluation provides the output that corresponds to a particular input, essentially telling you the value the function expects. This step is crucial in understanding how different inputs affect the outcome of the function, especially in exponential cases.