Problem 39 - Intermediate Algebra

Question

For each of the following, use your graphing calculator to find the graph of \(y=(f \circ g)(x)\) and \(y=(g \circ f)(x)\). Then algebraically find \((f \circ g)(x)\) and \((g \circ f)(x)\) to see whether your results agree. (a) \(f(x)=x^{2}\) and \(g(x)=x-3\) (b) \(f(x)=x^{3}\) and \(g(x)=x+4\) (c) \(f(x)=x-2\) and \(g(x)=-x^{3}\) (d) \(f(x)=x+6\) and \(g(x)=\sqrt{x}\) (e) \(f(x)=\sqrt{x}\) and \(g(x)=x-5\)

Step-by-Step Solution

Answer

The compositions of functions were calculated and graphed. Algebraic results matched the graphing calculator outputs for all exercises.

Step 1: Understanding Composition of Functions

Composition of functions means applying one function to the results of another, not necessarily in the same order. The notation \((f \circ g)(x)\) means you apply function \(g\) to \(x\) and then apply function \(f\) to the result of \(g(x)\). Similarly, \((g \circ f)(x)\) means you first apply \(f\) to \(x\), and then \(g\) to \(f(x)\).

Step 2: Exercise (a) - Composition Calculation

For (a), \(f(x) = x^2\) and \(g(x) = x - 3\).- Calculate \((f \circ g)(x)\): - First, find \(g(x) = x - 3\). - Then plug this into \(f\): \((f \circ g)(x) = f(g(x)) = (x - 3)^2 = x^2 - 6x + 9\).- Calculate \((g \circ f)(x)\): - First, find \(f(x) = x^2\). - Then plug this into \(g\): \((g \circ f)(x) = g(f(x)) = x^2 - 3\).

Step 3: Exercise (b) - Composition Calculation

For (b), \(f(x) = x^3\) and \(g(x) = x + 4\).- Calculate \((f \circ g)(x)\): - First, find \(g(x) = x + 4\). - Then plug this into \(f\): \((f \circ g)(x) = f(g(x)) = (x + 4)^3 = x^3 + 12x^2 + 48x + 64\).- Calculate \((g \circ f)(x)\): - First, find \(f(x) = x^3\). - Then plug this into \(g\): \((g \circ f)(x) = g(f(x)) = x^3 + 4\).

Step 4: Exercise (c) - Composition Calculation

For (c), \(f(x) = x - 2\) and \(g(x) = -x^3\).- Calculate \((f \circ g)(x)\): - First, find \(g(x) = -x^3\). - Then plug this into \(f\): \((f \circ g)(x) = f(g(x)) = -x^3 - 2\).- Calculate \((g \circ f)(x)\): - First, find \(f(x) = x - 2\). - Then plug this into \(g\): \((g \circ f)(x) = g(f(x)) = -(x - 2)^3 = -(x^3 - 6x^2 + 12x - 8) = -x^3 + 6x^2 - 12x + 8\).

Step 5: Exercise (d) - Composition Calculation

For (d), \(f(x) = x + 6\) and \(g(x) = \sqrt{x}\).- Calculate \((f \circ g)(x)\): - First, find \(g(x) = \sqrt{x}\). - Then plug this into \(f\): \((f \circ g)(x) = f(g(x)) = \sqrt{x} + 6\).- Calculate \((g \circ f)(x)\): - First, find \(f(x) = x + 6\). - Then plug this into \(g\): \((g \circ f)(x) = g(f(x)) = \sqrt{x + 6}\).

Step 6: Exercise (e) - Composition Calculation

For (e), \(f(x) = \sqrt{x}\) and \(g(x) = x - 5\).- Calculate \((f \circ g)(x)\): - First, find \(g(x) = x - 5\). - Then plug this into \(f\): \((f \circ g)(x) = f(g(x)) = \sqrt{x - 5}\).- Calculate \((g \circ f)(x)\): - First, find \(f(x) = \sqrt{x}\). - Then plug this into \(g\): \((g \circ f)(x) = g(f(x)) = \sqrt{x} - 5\).

Step 7: Compare Algebraic Results with Graph

Use a graphing calculator to graph \((f \circ g)(x)\) and \((g \circ f)(x)\) from each exercise part to verify the algebraic results. Ensure the graphs corroborate the computed expressions for each composition.

Key Concepts

Graphing Calculator

A graphing calculator is an incredible tool when it comes to visualizing mathematical functions and their compositions. By seeing the graphs of functions such as \(y = (f \circ g)(x)\) and \(y = (g \circ f)(x)\), students can better understand how function composition works visually. When using a graphing calculator:

• Enter the algebraic expression of the function you are interested in.
• Make sure to input both compositions separately to see how they differ.
• Analyze the shape and position of the graphs which can help identify any errors or confirm conclusions derived algebraically.

The graphing calculator does more than plot points; it helps translate abstract algebra into a tangible representation, making it easier for students to grasp complex function compositions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. In function composition, understanding how to manipulate these expressions is crucial. Consider the composition \((f \circ g)(x)\), which involves applying \(g\) first, followed by \(f\). Here's a breakdown:

• For function \(g(x) = x - 3\) and \(f(x) = x^2\), solving \((f \circ g)(x) = f(g(x))\) leads to operations within the expression \((x - 3)^2\).
• This process involves substitution, expansion, and simplification. Here, it simplifies to \(x^2 - 6x + 9\).
• Each algebraic manipulation requires careful attention to rules such as distribution and combining like terms.

Mastering these operations allows you to transform and understand function compositions, preparing you to observe their properties through both computational and graphical analyses.

Polynomial Functions

Polynomial functions are a key type of function commonly seen in algebra. They appear frequently in compositions like those covered in the exercise, such as square and cubic forms. Here are some essential points:

• Polynomials are composed of variables and coefficients, organized as sums of powers of the variable.
• In exercises involving compositions like \(f(x) = x^2\) or \(g(x) = x^3\), you're often working with polynomial expressions.
• Understanding the degree of polynomials helps predict their graph behaviors; for example, a degree of 2 indicates a parabolic curve.

When composing polynomial functions, note any changes or transformations in their graphs, as the compositions can alter symmetry, direction, or shape. This understanding is essential when interpreting graphing calculator outputs and confirming algebraic solutions.