Problem 38 - College Algebra

Question

Given the function \(f(x)=\sqrt{x+2}\) a. Evaluate \(f(7)\). b. Solve \(f(x)=4\)

Step-by-Step Solution

Answer

a. \( f(7) = 3 \). b. \( x = 14 \).

Step 1: Understand the Function

The given function is a square root function expressed as \( f(x) = \sqrt{x+2} \). This means for each input \( x \), the output is the square root of \( x+2 \).

Step 2: Evaluate \(f(7)\)

To find \( f(7) \), substitute \( x = 7 \) into the function: \[ f(7) = \sqrt{7+2} \] Simplify the expression: \[ f(7) = \sqrt{9} \] Since \( \sqrt{9} = 3 \), we have \( f(7) = 3 \).

Step 3: Set Up the Equation for \(f(x)=4\)

To solve \( f(x) = 4 \), set \( \sqrt{x+2} = 4 \) and solve for \( x \). This equation implies that the square root of \( x+2 \) should equal 4.

Step 4: Solve \(\sqrt{x+2} = 4\)

Square both sides of the equation to eliminate the square root: \[ (\sqrt{x+2})^2 = 4^2 \] This simplifies to: \[ x+2 = 16 \].

Step 5: Solve for \(x\) in the Equation \(x+2=16\)

Subtract 2 from both sides to isolate \( x \): \[ x = 16 - 2 \] which results in \( x = 14 \). So, when \( f(x) = 4 \), \( x = 14 \).

Key Concepts

Function Evaluation

The concept of function evaluation involves finding the output of a function for a given input. This is a fundamental skill in analyzing mathematical functions. For the function given in this exercise, \(f(x) = \sqrt{x+2}\), solving for \(f(7)\) requires substituting \(7\) into the function and evaluating:

• First, substitute \(x = 7\) into the function to get \(f(7) = \sqrt{7+2}\).
• Simplify the expression: \(f(7) = \sqrt{9}\).
• Find the square root: since \(\sqrt{9} = 3\), thus \(f(7) = 3\).

When evaluating functions, it's important to follow these steps carefully. Always substitute correctly and simplify the expression step by step. Knowing how to evaluate a function allows us to understand how the function behaves for specific inputs.

Solving Equations

Solving equations often requires finding the value of the variable that makes the equation true. When solving \(f(x) = 4\) for the function \(f(x) = \sqrt{x+2}\), we set up the equation such that \(\sqrt{x+2} = 4\). Here is the step-by-step process:

• Set the equation: \(\sqrt{x+2} = 4\).
• Square both sides to eliminate the square root: \((\sqrt{x+2})^2 = 4^2\).
• This simplifies to: \(x + 2 = 16\).
• Solve for \(x\): Subtract 2 from both sides, \(x = 16 - 2\).
• Thus, \(x = 14\).

By following these steps, you can solve many types of equations. Always isolate the variable by performing reversible operations, ensuring that each step maintains equality in the equation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators. In the function \(f(x) = \sqrt{x+2}\), the expression \(x+2\) inside the square root is an algebraic expression. Understanding algebraic expressions is crucial for evaluating and solving functions and equations:

• An algebraic expression, like \(x+2\), can be evaluated by substituting specific values for variables.
• In this exercise, we substitute \(x = 7\) to evaluate the function and simplify: \(x+2\) becomes \(9\).
• When solving equations, the goal may involve manipulating these expressions to isolate the variable.

Evaluating and manipulating algebraic expressions are essential skills for success in algebra. They allow us to translate problems into mathematical language that we can solve or simplify.