Problem 36 - Algebra and Trigonometry

Question

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=\sqrt{x+2} $$

Step-by-Step Solution

Answer

Domain: \([-2, \infty)\); Range: \([0, \infty)\).

Step 1: Understanding the Function

The function given is \( f(x) = \sqrt{x + 2} \). This is a square root function, which means the expression under the square root, \(x + 2\), must be greater than or equal to zero to avoid complex numbers.

Step 2: Finding the Domain

To find the domain, solve the inequality \( x + 2 \geq 0 \). Solve for \( x \):\[ x + 2 \geq 0 \] Subtract 2 from both sides: \[ x \geq -2 \] Thus, the domain of \( f(x) \) is all real numbers \( x \) such that \( x \geq -2 \), which can be written as \([-2, \infty)\).

Step 3: Graphing the Function

Using a graphing calculator, plot the function \( f(x) = \sqrt{x + 2} \). This graph starts from the point (-2, 0) since the square root becomes 0 when \( x = -2 \), and it increases as \( x \) increases.

Step 4: Determining the Range

Based on the graph, the smallest value of \( f(x) \) occurs at \( x = -2 \), which is \( f(-2) = \sqrt{-2 + 2} = 0 \). As \( x \) increases, \( f(x) \) increases without bound. Thus, the range is \([0, \infty)\).

Key Concepts

Understanding Domain and Range

In the function \( f(x) = \sqrt{x+2} \), the domain and range are key concepts to grasp. The **domain** refers to all the possible input values \( x \) that make the function work, i.e., not resulting in complex numbers.
In square root functions like this, you must ensure that the expression under the root is non-negative. Solve the inequality \( x + 2 \geq 0 \) to find that \( x \geq -2 \). Thus, the domain is all real numbers greater than or equal to -2, written as \([-2, \infty)\).
The **range** instead is about output values. Since the smallest output happens at \( x = -2 \) where \( f(x) = 0 \) and grows as \( x \) increases, the range is \([0, \infty)\). This tells you \( f(x) \) starts at zero and increases without limit.

Using a Graphing Calculator

A graphing calculator can greatly simplify understanding the function \( f(x) = \sqrt{x+2} \). By plotting the function, you gain a visual representation that helps you see the function's behavior through its graph. To achieve this:

• Input the function \( \sqrt{x+2} \) into the calculator's graphing section.
• Set an appropriate scale to include the domain \( x \geq -2 \).
• Observe the graph forming a curve starting from the point \((-2, 0)\), moving upwards as it stretches endlessly to the right.

Graphing shows the starting point and how the curve moves upwards, reinforcing the understanding of the domain and range. This visual aid is useful for solving problems and predicting values just by looking.

Examining Inequalities in Functions

Understanding inequalities in functions like \( f(x) = \sqrt{x+2} \) involves recognizing which values satisfy the inequality. Given that square roots must be non-negative, solve for \( x \) by working with inequalities. It's as follows:

• Consider the expression under the square root: \( x+2 \).
• The inequality is \( x + 2 \geq 0 \), ensuring no negative roots.
• Subtract 2 from both sides, yielding \( x \geq -2 \).

This shows the permissible input values—forming the domain. Inequalities also help understand how functions will behave in different scenarios.
When graphed, these inequalities assure the line starts right where \( x = -2 \), and values stretch infinitely forward, making the math both visual and intuitive.