Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find \(f \circ g, g \circ f,\) and the domain of \(f \circ g .\) (b) Use a graphing utility to graph \(f \circ g\) and \(g \circ f .\) Determine whether \(f \circ g=g \circ f.\) $$f(x)=|x|, \quad g(x)=-x^{2}+1$$

Step-by-Step Solution

Answer

The composition \(f(g(x)) = |-x^{2} + 1|\), \(g(f(x)) = - |x|^{2} + 1\), and the domain of \(f \circ g\) is all real numbers. By plotting the graphs, \(f \circ g\) is not equal to \(g \circ f\). They have different shapes, thus they are not equal.

Step 1: Find \(f \circ g\)

The composition \(f \circ g\) means applying function \(f\) to \(g(x)\). Here, \(f(g(x)) = f(-x^{2} + 1) = |-x^{2} + 1|\.

Step 2: Find \(g \circ f\)

The composition \(g \circ f\) means applying function \(g\) to \(f(x)\). Here, \(g(f(x)) = g(|x|) = - |x|^{2} + 1\.

Step 3: Find the domain of \(f \circ g\)

The domain of a function is the set of all values for which the function is defined. Here, \(f \circ g\) is defined for all real numbers because the domain of \(f(x) = |x|\) is all real numbers and the domain of \(g(x) = -x^{2} + 1\) is also all real numbers. So, the domain of \(f \circ g\) is all real numbers.

Step 4: Graph \(f \circ g\) and \(g \circ f\)

Use any graphing utility (like GeoGebra, Desmos, or a graphing calculator) to graph the curves of \(f \circ g = |-x^{2} + 1|\) and \(g \circ f = - |x|^{2} + 1\).

Step 5: Compare \(f \circ g\) and \(g \circ f\)

Now compare the graphs of \(f \circ g\) and \(g \circ f\). If both graphs are the same, then \(f \circ g = g \circ f\). If not, they are not equal.

Key Concepts

Absolute Value

The absolute value of a number is a way to describe how far the number is from zero on the number line, symbolized by vertical bars like \( |x| \) for a number \( x \). It essentially gives us the magnitude of a number without considering its sign. For any real number, if the number is positive or zero, its absolute value is the same as the number itself. However, if the number is negative, it is made positive. Here are the basics of absolute value:

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

In the context of the function \( f(x) = |x| \), the absolute value affects any input of \( x \) by making it non-negative. When composing functions, the absolute value ensures that the output will always be a non-negative outcome, regardless of the sign of the input it receives. This characteristic makes it distinctive in function operations and graph interpretations, making real understanding of absolute value crucial when dealing with function compositions like \( f(g(x)) \).

Quadratic Function

A quadratic function is expressed in the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a \) is not equal to zero. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the leading coefficient \( a \).In this exercise, we have \( g(x) = -x^2 + 1 \). Due to the negative sign in front of the \( x^2 \) term, \( g(x)\) opens downwards with a vertex at \( (0, 1) \), making this parabola upside-down. Here are some quick properties of quadratic functions:

• A positive \( a \) results in a parabola opening upwards.
• A negative \( a \) flips it to open downwards.
• The vertex form of a quadratic can give us the maximum or minimum point of the function easily.

When we compose this quadratic function \( g(x)\) with another, such as with the absolute value function in \( f(g(x)) = |-x^2 + 1| \), we alter not only the graph's direction but also the way it's interpreted, as the absolute value ensures the entire graph remains non-negative. This insights helps deeply while graphing combinations like \( f \) composed \( g \).

Domain of a Function

The domain of a function is the complete set of possible input values (or \( x \) values) for which the function is defined. For a basic understanding, any real number can be input into the function if it does not make the function undefined (e.g., dividing by zero or taking the square root of a negative number in real numbers).In our exercise, \( f \) with \( f(x) = |x| \) and \( g \) with \( g(x) = -x^2 + 1 \) are examined for their domain. Both functions here do not have any restrictions like division by zero or square roots of negative numbers, which are common culprits for undefined conditions. Specifically:

• The absolute value function \( f(x) = |x| \) is defined for all real numbers since any real number can be made non-negative.
• The quadratic function \( g(x) = -x^2 + 1 \) is a basic polynomial, also defined for all real numbers.

Therefore, the composition \( f(g(x)) \) maintains its definition over all real numbers because there are no additional restrictions introduced. Understanding the domain is key, especially when graphing \( f \) and \( g \) as compositions; knowing their domain ensures you correctly identify where the graph exists on the coordinate plane.