Problem 25 - Algebra 2

Question

Graph each function. Identify the domain and range. \(f(x)=|x+2|\)

Step-by-Step Solution

Answer

Domain: all real numbers \((\mathbb{R})\). Range: \([0, \infty)\).

Step 1: Understanding the Function

The function given is the absolute value function \(f(x) = |x+2|\). This represents a V-shaped graph that opens upwards with its vertex at the point where the expression inside the absolute value is zero.

Step 2: Finding the Vertex

To find the vertex of the function, set the inside of the absolute value equal to zero: \(x+2=0\). Solving this gives \(x=-2\). Thus, the vertex of the graph is \((-2, 0)\).

Step 3: Determining the Domain

The domain of an absolute value function of the form \(f(x) = |x+2|\) is all real numbers \(\mathbb{R}\). This is because the expression inside the absolute value is defined for all real values of \(x\).

Step 4: Determining the Range

The range of \(f(x) = |x+2|\) is \([0, \infty)\). This is because the absolute value function outputs only non-negative values, starting from zero (at the vertex) and extending to infinity.

Step 5: Sketching the Graph

To graph \(f(x) = |x+2|\), plot the vertex at \((-2, 0)\). Plot additional points by choosing values for \(x\) and computing \(f(x)\): for example, \(f(-3) = |-1| = 1\) and \(f(-1) = |1| = 1\). Connect these points with a V-shape.

Key Concepts

Vertex

The vertex of an absolute value function is an essential piece of information because it helps us understand the point at which the function achieves its minimum value, especially for functions like \(f(x) = |x+2|\).The vertex is the point where the direction of the graph changes, creating a V-shape.

For the function given, the inside of the absolute value is \(x+2\). To find the vertex, we set this equal to zero:

• \(x+2 = 0\)
• Solving gives \(x = -2\)

So, the vertex is located at the point \((-2, 0)\).

This is the point where the graph reaches its lowest value before increasing again in both directions.The vertex forms the base of the V-shape, indicating the minimum output of the function.

Domain

The domain of a function is all the possible values of \(x\) for which the function is defined. For absolute value functions, like our example \(f(x) = |x+2|\), the domain is generally vast because absolute values are defined for any real number.

To put it simply, there are no restrictions on the values \(x\) can take:

• The expression \(x+2\) inside the absolute value doesn't cause any mathematical issues like division by zero or taking the square root of a negative number.
• This means the function is defined for all real numbers, \(\mathbb{R}\).

So, you can plug any real number into this function and get a valid result in return. The whole number line is open for use!

Range

The range of a function refers to all the possible output values it can produce. For absolute value functions like \(f(x) = |x+2|\), the range is often restricted to non-negative numbers.

Why? Because absolute values always give us non-negative results – they measure the distance from zero, disregarding whether the input was negative or positive.

• The smallest value \(f(x)\) can take is at the vertex, where \(x = -2\) and \(f(x) = 0\).
• After this point, as \(x\) moves in either direction, \(f(x)\) increases, heading towards infinity.

Therefore, the range for this function is \([0, \infty)\), starting at zero and extending upwards indefinitely.