Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-8,\) for each function: a. Write an expression for \(f(x) .\) b. Find \(f(5)\) \(y=\sqrt{x-1}\)

Step-by-Step Solution

Answer

\( f(x) = \sqrt{x-1} \); \( f(5) = 2 \).

Step 1: Identify the function

The given function is provided as \( y = \sqrt{x-1} \). This means that \( f(x) = \sqrt{x-1} \).

Step 2: Substitute x-value into the function

To find \( f(5) \), substitute \( x = 5 \) into the function. This gives us \( f(5) = \sqrt{5-1} \).

Step 3: Simplify the expression

Simplify the expression within the square root: \( 5-1 = 4 \). Thus, \( f(5) = \sqrt{4} \).

Step 4: Calculate the square root

The square root of 4 is 2, so \( f(5) = \sqrt{4} = 2 \).

Key Concepts

Square Root Function

The square root function is a fundamental concept in mathematics. It's often represented by the square root symbol \( \sqrt{} \). This function is used to determine a number which, when multiplied by itself, gives the original number inside the square root symbol.
If you're looking at the function \( y = \sqrt{x-1} \), it indicates that we take the square root of the expression \( x-1 \). The result, \( y \), is the value produced when the expression inside the square root is calculated. This function is crucial for solving problems involving calculations where finding the base number is important. Square root functions appear in various forms across algebra and calculus.

Substitution Method

The substitution method is a common technique used in function evaluation and solving equations. It involves replacing a variable with a specific value to simplify the function. In this exercise, we're asked to find \( f(5) \) for the function \( f(x) = \sqrt{x-1} \).

By substituting \( x = 5 \) into the function, we replace the variable \( x \) with 5, leading to \( f(5) = \sqrt{5-1} \). This process simplifies the equation into a calculation that can be completed step by step. Substitution not only helps in evaluating functions but also in finding roots and solutions to algebraic equations. It's an efficient way to bring clarity and concrete numbers into mathematical expressions.

Simplifying Expressions

Simplifying expressions is a vital skill in mathematics as it makes complex calculations more manageable. It involves reducing a mathematical expression into its simplest form. In our given problem, after substituting \( x = 5 \) in \( \sqrt{x-1} \), we have the expression \( \sqrt{5-1} \).

By simplifying this, we first perform the operation inside the parentheses: \( 5-1 \), which results in \( 4 \). Thus, our expression becomes \( \sqrt{4} \). Simplifying expressions helps in minimizing errors and finding solutions more swiftly during mathematical problem-solving. It's like tidying up the workspace before diving deep into problem resolution.

Calculating Square Roots

Calculating square roots is the final step in evaluating square root functions. Once you've simplified an expression to \( \sqrt{4} \), the next step is to find the square root of 4.

Finding a square root is determining which number, when multiplied by itself, equals the given number. For \( \sqrt{4} \), we ask "What number times itself gives 4?" The answer is 2, because \( 2 \times 2 = 4 \). Hence, \( \sqrt{4} = 2 \).

This concept of calculating square roots is fundamental and applicable across many areas of math, including geometry and algebra. Understanding square roots allows for greater comprehension of how numbers relate to each other in the context of multiplication and area calculations.