Problem 7 - Algebra A Combined Function

Question

Sketch the graph of each function. $$ f(x)=\sqrt{x+2} $$

Step-by-Step Solution

Answer

Sketch a graph that begins at (-2,0), curves upward through (0,√2), and continues through (2,2), only existing for \( x \geq -2 \).

Step 1: Understand the Function

The function given is \( f(x) = \sqrt{x+2} \). This is a square root function, which generally has a shape like half of a parabola. Its domain is determined by the expression inside the square root being greater than or equal to zero.

Step 2: Determine the Domain

Set the expression inside the square root \( x + 2 \) greater than or equal to zero: \( x + 2 \geq 0 \). Solve this inequality to find \( x \geq -2 \). Thus, the domain of the function is \( x \geq -2 \).

Step 3: Find the Key Points

Calculate key points of the function by substituting values of \( x \) from its domain into \( f(x) \). For example:- When \( x = -2 \), \( f(-2) = \sqrt{-2 + 2} = 0 \).- When \( x = 0 \), \( f(0) = \sqrt{0 + 2} = \sqrt{2} \approx 1.41 \).- When \( x = 2 \), \( f(2) = \sqrt{2 + 2} = \sqrt{4} = 2 \).These points are \((-2, 0)\), \((0, \sqrt{2})\), and \((2, 2)\).

Step 4: Sketch the Graph

Plot the points \((-2, 0)\), \((0, \sqrt{2})\), and \((2, 2)\) on a coordinate plane. Since the function is a square root, it will start at \((-2, 0)\) and increase slowly at first and then more rapidly in a curve that resembles half of a parabola opening to the right. Draw a smooth curve through these points.

Step 5: Review the Characteristics

Make sure that the graph starts at \( x = -2 \) and continues to the right, confirming that it does not exist for \( x < -2 \). The graph should correctly represent the increasing behavior of the function and match the calculated key points.

Key Concepts

Square Root Function

The square root function is a fundamental concept in mathematics. It generally takes the form \( f(x) = \sqrt{x} \) or variants like \( f(x) = \sqrt{x+c} \), where the graph usually resembles half of a parabola. This is because square root functions only produce non-negative outputs.
One notable characteristic of square root functions is their behavior. As the value of \( x \) increases, so does the value of \( f(x) \), but at a decreasing rate. This results in a graph that starts with a steep slope and then levels off, forming a gentle curve. For a function like \( f(x) = \sqrt{x+2} \), the "+2" shifts the entire graph two units to the left. As a result, the graph starts at a different point on the x-axis, specifically, at \( x = -2 \). Understanding this shift is essential when sketching or working with square root functions.

Domain and Range

Determining domain and range is crucial for understanding any function, including square root functions. The domain of a function is the complete set of possible values of \( x \) that make the function work and produce real numbers. For the square root function \( f(x) = \sqrt{x+2} \), the expression inside the square root must be non-negative.
This means solving the inequality \( x + 2 \geq 0 \), which gives us \( x \geq -2 \). Hence, the domain of this function is all real numbers greater than or equal to -2.

• The domain: \( x \geq -2 \)
• The range: \( f(x) \geq 0 \)

The range, on the other hand, refers to all possible output values of the function. Since the square root function only outputs non-negative values, the range of \( f(x) = \sqrt{x+2} \) is all real numbers \( f(x) \geq 0 \).

Plotting Key Points

Plotting key points of a function is essential in accurately graphing it. It involves choosing values of \( x \) from the domain and finding the corresponding \( f(x) \) values.
For the function \( f(x) = \sqrt{x+2} \), start by determining points that are easy to compute, such as:

• When \( x = -2 \), \( f(-2) = \sqrt{-2 + 2} = 0 \), giving the point \((-2, 0)\).
• When \( x = 0 \), \( f(0) = \sqrt{0 + 2} = \sqrt{2} \), roughly \(1.41\), providing the point \((0, \sqrt{2})\).
• When \( x = 2 \), \( f(2) = \sqrt{2 + 2} = 2 \), resulting in the point \((2, 2)\).

These points illustrate how the function behaves and where its curve is plotted on the graph.
By marking these points on a coordinate axes and drawing a smooth curve through them, we can visualize the shape of the function effectively.

Solving Inequalities

Understanding how to solve inequalities is a key skill when dealing with functions, particularly when determining the domain. An inequality shows the relationship between expressions that are not equal, using symbols like \(>\), \(<\), \(\geq\), or \(\leq\).
In the context of the square root function \( f(x) = \sqrt{x+2} \), you must solve \( x + 2 \geq 0 \) to find valid \( x \) values. This process is:

• Subtract 2 from both sides: \( x \geq -2 \)

Understanding inequality expressions ensures that you obtain the appropriate domain for the function.
Solving inequalities helps define key aspects of functions and allows you to understand where they are defined, enhancing your graphing and analytical skills.