Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). Given the function \(f(x)=\sqrt{x+2}\) (a) Evaluate \(f(7)\). (b) Solve \(f(x)=4\).

Step-by-Step Solution

Answer

(a) 3; (b) x = 14.

Step 1: Evaluate f(7) using the function

We are given the function \( f(x) = \sqrt{x + 2} \) and need to find \( f(7) \). First, substitute \( x = 7 \) into the function: \[\begin{align*}f(7) &= \sqrt{7 + 2} \ &= \sqrt{9} \ &= 3.\end{align*}\] Thus, \( f(7) = 3 \).

Step 2: Solve f(x)=4 for x

Next, we solve the equation \( f(x) = 4 \) where \( f(x) = \sqrt{x + 2} \). Set the equation \( \sqrt{x + 2} = 4 \) and solve for \( x \). Square both sides to eliminate the square root: \[ \begin{align*}(\sqrt{x + 2})^2 &= 4^2 \ x + 2 &= 16.\end{align*}\]Subtract 2 from both sides to solve for \( x \): \[ x = 16 - 2 = 14. \]Thus, \( x = 14 \).

Key Concepts

Function Evaluation

Function evaluation is all about finding the value of a function at a specific point. It's like substituting the number into the equation wherever you see the variable. For example, if we need to evaluate the function \( f(x) = \sqrt{x+2} \) at \( x = 7 \), we replace \( x \) with 7 in the equation.
Here's the simple process for evaluating a function:

• Identify the function and the value of the variable you want to use. For instance, \( f(x) = \sqrt{x+2} \) and \( x = 7 \).
• Substitute the number (7) into the function wherever you see \( x \).
• Perform the arithmetic operation: Calculate \( \sqrt{7+2} = \sqrt{9} = 3 \)

So, the value of \( f(7) \) is 3. Function evaluation is a core concept in mathematics that helps to understand how functions behave for different inputs. You practice these steps, and soon you'll be able to do them with ease!

Square Root Functions

Square root functions are equations that involve a square root operation. In simple terms, a square root function is one that contains an expression under the square root symbol, like \( \sqrt{x+2} \). In the world of mathematics, square root functions are a staple, appearing everywhere from geometry to statistics. When working with square root functions, there are a few things to keep in mind:

• The expression inside the square root (the radicand) must be non-negative because you cannot take the square root of a negative number in real numbers.
• The output of a square root function is always non-negative.

For the function \( f(x) = \sqrt{x+2} \), as long as \( x+2 \geq 0 \), finding \( f(x) \) is straightforward: just find the square root of \( x+2 \).
These functions are great for modeling real-world phenomena, like calculating the dimensions of a square when given its area. Being comfortable with how they work is an essential skill in any mathematician's toolkit. Once mastered, they can be incredibly useful.

Solving Equations Involving Functions

Solving equations involving functions means finding values that make a function true. It often involves isolating a variable to solve an equation. Suppose we have an equation like \( f(x) = 4 \), where \( f(x) = \sqrt{x+2} \). Here's how you can solve this equation step-by-step:

• Start by setting the function equal to the given value: \( \sqrt{x+2} = 4 \).
• To eliminate the square root, square both sides of the equation. It becomes \( (\sqrt{x+2})^2 = 4^2 \), which simplifies to \( x+2 = 16 \).
• Now, isolate the variable \( x \). Subtract 2 from both sides to get \( x = 14 \).

Solving such equations is akin to unraveling a puzzle, piece by piece. It's essential to methodically handle each part of the equation to find the correct solution. Practice solving equations involving functions sharpen your problem-solving skills and mathematical intuition.