Problem 20 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=|x| .\) Graph the functions on the same set of axes. \(g(x)=-(x+2)^{2}+1\) and \(f(g(x))\)

Step-by-Step Solution

Answer

The first graph \(g(x)\) is an inverted parabola shifted 2 units to the left and 1 unit up. The second graph \(f(g(x))\) is the absolute value of \(g(x)\), thus every negative y-value of the \(g(x)\) graph is reflected above the x-axis, creating a 'V' shaped graph, mirroring positive y-values of \(g(x)\) across the x-axis.

Step 1: Graph Function g(x)

Firstly, graph the function \(g(x) = -(x+2)^2+1\). The basic shape of \(g(x)\) is an inverted parabola that has been shifted 2 units to the left and 1 unit upward.

Step 2: Find the Composition of Function f and g

The composition of function \(f\) and \(g\), expressed as \(f(g(x))\), involves applying the function \(f\) to the result of \(g(x)\). Since function \(f(x) = |x|\) and \(g(x) = -(x+2)^2 + 1\), composition \(f(g(x))\) will be: \(f(g(x)) = |-(x+2)^2 + 1|\).

Step 3: Plot f(g(x)) on the Graph

Plot the \(f(g(x))\) function on the same set of axes with \(g(x)\). This will result in a graph showing two functions on same set of axes where \(f(g(x))\) will have a similar shape as \(g(x)\) except that it will reflect all the points of \(g(x)\) which are below the x-axis, to above the x-axis.

Key Concepts

Absolute Value Function

The absolute value function, denoted as \( f(x) = |x| \), is a fundamental concept in mathematics. It is defined as the non-negative value of \( x \). This means that the absolute value of a number is its distance from zero on the number line, regardless of direction.

• If \( x \) is positive or zero, \( |x| = x \).
• If \( x \) is negative, \( |x| = -x \), which makes it positive.

The graph of the absolute value function is V-shaped.

It has a vertex at the origin (0,0). The graph consists of two linear pieces: one that rises as it moves to the right and one that rises as it moves to the left of the vertex.

When applied as \( f(g(x)) \), like in our exercise, it transforms the output of the function \( g(x) \) to be all non-negative values. In essence, the absolute value composition reflects any negative parts of the graph of \( g(x) \) to its positive counterpart across the x-axis.

Parabola Transformations

Parabola transformations involve altering the basic form of a parabola, generally expressed as \( y = ax^2 + bx + c \), to shift, stretch, compress, or flip it.

For instance, consider the transformation in the function \( g(x) = -(x+2)^2 + 1 \):

• The negative sign in front of \( (x+2)^2 \) flips the parabola upside down, indicating that it opens downward.
• The term \((x+2)^2\) shifts the parabola 2 units to the left. This is known as a horizontal shift and occurs because of the \(+2\) inside the bracket.
• The \(+1\) at the end of the equation is a vertical shift that moves the entire parabola 1 unit upward.

These transformations are crucial for graphing accurately, as they specify the vertex's place and direction in which the parabola opens, thus defining its overall shape and orientation in the graph.

Graphing Functions

Graphing functions is a fundamental skill in understanding their behavior visually. By plotting equations on a coordinate grid, we can observe relationships and patterns.

To graph functions like \( g(x) = -(x+2)^2 + 1 \) and its composition \( f(g(x)) \), follow these steps:

• Identify and mark the transformations on the coordinate grid. For \( g(x) \), these include the left shift, downward opening, and upward shift mentioned previously.
• Plot the vertex, which in the case of \( g(x) \), is at \((-2, 1)\).
• Sketch the parabola, ensuring that it's opening downward, and notice how the graph reflects all points below the x-axis when applying \( f(g(x)) \).
• For \( f(g(x)) \), use the absolute value to reflect the graph so all values are non-negative, ensuring its similarities and differences from \( g(x) \) are visible on the same axes.

Graphing helps uncover the full impact of function compositions and transformations, providing insights not just on their algebraic properties but also on their geometric manifestations.