Problem 67 - Elementary and Intermediate Algebra

Question

Use a calculator to find each function value. Round to the nearest ten- thousandth. See Example 5 and Using Your Calculator \(f(x)=\sqrt{x^{2}+1}\) a. \(f(4)\) b. \(f(2.35)\)

Step-by-Step Solution

Answer

\( f(4) = 4.1231 \) and \( f(2.35) = 2.5534 \).

Step 1: Understanding the Function

The function given is \( f(x) = \sqrt{x^2 + 1} \). This function requires us to first square the input \( x \), then add 1 to it, and finally take the square root of the result.

Step 2: Calculating \( f(4) \)

For part (a), we substitute \( x = 4 \) into the function:1. Calculate \( 4^2 = 16 \).2. Add 1 to the result: \( 16 + 1 = 17 \).3. Take the square root of 17. Using a calculator, \( \sqrt{17} \approx 4.1231 \).So, \( f(4) = 4.1231 \).

Step 3: Calculating \( f(2.35) \)

For part (b), we substitute \( x = 2.35 \) into the function:1. Calculate \( (2.35)^2 = 5.5225 \).2. Add 1 to the result: \( 5.5225 + 1 = 6.5225 \).3. Take the square root of 6.5225. Using a calculator, \( \sqrt{6.5225} \approx 2.5534 \).So, \( f(2.35) = 2.5534 \).

Key Concepts

Function Evaluation

Function evaluation is a key aspect of algebra, and it involves determining the output of a function for a given input. For the function \( f(x) = \sqrt{x^2 + 1} \), you start by replacing \( x \) with the input value you are evaluating. For instance, if you need to find \( f(4) \), you replace \( x \) with 4. Here is how you would go about it:

• Square the input number.
• Add one to the result.
• Take the square root of the new sum.

The process essentially transforms your input number through a defined rule, resulting in an output value. Each step changes the number according to the function's formula, illustrating how a simple input can create a complex result.
Function evaluation helps us predict outcomes and solve problems that utilize these specific rules. Understanding how each step works ensures clearer insight into what a function does, making it a powerful tool in algebra.

Square Roots

Square roots are a fundamental concept in mathematics, often appearing in functions and equations. A square root of a number \( y \) is a number \( x \) such that \( x^2 = y \). It's symbolized by the radical sign (\(\sqrt{}\)). Square roots are critical when dealing with quadratic expressions and various algebraic computations.
When evaluating a square root in the context of a function like \( f(x) = \sqrt{x^2 + 1} \), you first perform the operations inside the square root, then simplify as needed. If we look at an example such as finding the square root of 17, it implies you're seeking a number which, when squared, gives 17. Using your calculator helps here because many square roots don't resolve neatly to whole numbers.

• The number you're rooting affects the precision needed.
• Often, calculators give approximate values for square roots.
• Square roots are essential in understanding curves and roots of polynomial functions.

Grasping the role of square roots is crucial in mathematics as it facilitates working through complex equations and understanding their behavior.

Using a Calculator in Mathematics

Calculators are indispensable tools in mathematics for dealing with complex calculations quickly and accurately. They come especially handy in evaluating functions that require non-trivial computations, such as square roots, exponentiations, and arithmetic operations.
In the function \( f(x) = \sqrt{x^2 + 1} \), after simplifying within the square root, you might need a calculator to obtain the precise value, especially when the square root results in an irrational number.
Here's how calculators can assist:

• Ensure all steps are entered as instructed to get accurate results.
• Use the appropriate mode (scientific) to handle complex calculations.
• Enter calculations in stages if they become too complicated, such as first squaring, then summing, and finally rooting.

Calculators don't just speed up the process but also reduce the chance of human error, aiding significantly in ensuring output precision. Coupled with manual computation skills, calculators make learning and applying algebraic concepts much more accessible.