Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=|x+2|\)

Step-by-Step Solution

Answer

The graph of the function \(f(x)=|x+2|\) is a V-shape with the vertex at (-2,0). The function is neither even nor odd.

Step 1: Graphing the function

To begin with, understand that the graph of \(f(x) = |x|\) is a V-shaped plot intersecting the axis at the origin. Now, \(f(x) = |x+2|\) is a shift of the base function \(f(x) = |x|\) to the left by two units. Thus, the vertex of the V-shape will be at (-2,0). The line will go upwards to the left and to the right as the value of \(x\) increases or decreases.

Step 2: Testing for evenness

Substitute \(x\) with \(-x\) in \(f(x)\). We get \(f(-x) = |-x+2|\). Now, |-x+2| = |x-2|. Clearly, \(f(-x) \neq f(x)\), therefore \(f(x)\) is not even.

Step 3: Testing for oddness

The negation of \(f(x)= |x+2|\) is \(-f(x)=-|x+2|\). When we substitute \(x\) with \(-x\) in \(-f(x)\), we get \(-f(-x)=-|-(-x)+2|\) which is \(-f(-x)=-|-x+2|\). Here again, |-x+2| is not equal to |-x-2|. Thus, \(f(-x) \neq -f(x)\), therefore \(f(x)\) is not odd.

Key Concepts

Even and Odd Functions

Understanding the distinction between even and odd functions is crucial for students delving into the world of algebra and precalculus. An even function is symmetrical along the y-axis, which means that for every point on the graph, there is a mirrored point on the opposite side of the y-axis. Mathematically, a function f(x) is even if for every x in the domain, f(x) = f(-x).

Odd functions, on the other hand, have point symmetry about the origin. In other words, rotating the graph 180 degrees about the origin brings the function onto itself. In algebraic terms, f(x) is odd if -f(x) = f(-x) for all x in the domain. Clear examples of even and odd functions are f(x) = x^2 (even) and f(x) = x^3 (odd).

In the exercise, we tested the function f(x) = |x+2| and found that it is neither even nor odd. The properties of symmetry did not hold when we substituted x with -x, which means the graph does not mirror along the y-axis nor does it have rotational symmetry about the origin.

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, and reflecting the graph of a base function. Every transformation corresponds to a particular algebraic manipulation of the function's formula. The most basic transformations include:

• Vertical shifts which are achieved by adding or subtracting a value to f(x), moving the graph up or down.
• Horizontal shifts occur by adding or subtracting a value inside the function's argument, shifting the graph left or right.
• Vertical stretches and compressions are caused by multiplying f(x) by a factor, stretching or compressing the graph along the y-axis.
• Horizontal stretches and compressions occur when the input is multiplied by a factor, altering the graph along the x-axis.
• Reflections flip the graph over the x-axis or y-axis, depending on whether the function or its argument is multiplied by -1.

For the absolute value function in the exercise, f(x) = |x+2| represents a horizontal shift of 2 units to the left from the parent function f(x) = |x|. This transformation moves the entire graph, including the vertex of the 'V' shape, along the horizontal axis without altering its overall shape.

Absolute Value Equations

Solving absolute value equations is a key skill in algebra. An absolute value represents the distance of a number from zero on the number line, and it is always non-negative. The absolute value of x is written as |x|.

When an equation includes an absolute value term, it will have two possible solutions because both the positive and negative versions of the value yield the same absolute value. For instance, the equation |x| = 3 has two solutions: x = 3 and x = -3.

In graphing terms, the equation f(x) = |g(x)| creates a V-shaped graph, where g(x) dictates the form of the V. To graph an absolute value equation, we often start by identifying its vertex (the point at the bottom or top of the V) and then sketch the linear pieces on either side of the vertex.

The exercise provided asks students to graph f(x) = |x+2|, a simple absolute value equation, where the graph is a V-shaped curve with the vertex at the point (-2, 0). Solving absolute value equations and graphing them are foundational skills that help in understanding more complex functions and equations.