Problem 46 - Algebra for College Students

Question

For each of the following, find the graph of \(y=\) \((f \circ g)(x)\) and of \(y=(g \circ f)(x)\). (a) \(f(x)=x^{2}\) and \(g(x)=x+5\) (b) \(f(x)=x^{3}\) and \(g(x)=x+3\) (c) \(f(x)=x-6\) and \(g(x)=-x^{3}\) (d) \(f(x)=x^{2}-4\) and \(g(x)=\sqrt{x}\) (e) \(f(x)=\sqrt{x}\) and \(g(x)=x^{2}+4\) (f) \(f(x)=\sqrt[3]{x}\) and \(g(x)=x^{3}-5\)

Step-by-Step Solution

Answer

Compose functions, calculate \((f \circ g)(x)\) and \((g \circ f)(x)\), and graph them.

Step 1: Understand Composition Functions

Function composition involves applying one function to the results of another function. For example, if we first apply function \(g\) to \(x\) and then apply function \(f\) to the result of \(g(x)\), we get \((f \circ g)(x) = f(g(x))\). Conversely, \((g \circ f)(x) = g(f(x))\) means applying \(f\) first and then \(g\).

Step 2: Solve for (f ∘ g)(x) = f(g(x))

For each pair of functions, compute \((f \circ g)(x)\):- (a) \(f(g(x)) = ((x + 5)^2)\).- (b) \(f(g(x)) = ((x + 3)^3)\).- (c) \(f(g(x)) = (-x^3 - 6)\).- (d) \(f(g(x)) = ((\sqrt{x})^2 - 4)\).- (e) \(f(g(x)) = (\sqrt{x^2+4})\).- (f) \(f(g(x)) = \sqrt[3]{x^3-5}\) after simplification.

Step 3: Solve for (g ∘ f)(x) = g(f(x))

For each pair:- (a) \(g(f(x)) = (x^2 + 5)\).- (b) \(g(f(x)) = (x^3 + 3)\).- (c) \(g(f(x)) = -(x-6)^3\).- (d) \(g(f(x)) = \sqrt{x^2 - 4}\) with \(y\) being a real number for \(x \geq 2\) or \(x \leq -2\).- (e) \(g(f(x)) = ((\sqrt{x})^2 + 4)\), simplifying gives \(x + 4\).- (f) \(g(f(x)) = (\sqrt[3]{x})^3 - 5\), simplifying gives \(x - 5\).

Step 4: Graph Each Function

Using the above expressions, for each part, graph both \((f \circ g)(x)\) and \((g \circ f)(x)\) on the same set of axes. Consider the domain and range restrictions, especially where square roots and cube roots are involved. Note that compositions can alter these domains.

Key Concepts

Polynomial Functions

Polynomial functions are vital building blocks in mathematics. They consist of terms made up of constants, variables, and exponents that are non-negative integers. Here’s a breakdown of their characteristics:

• A polynomial function of degree 2 is called a quadratic, like \(f(x) = x^2\), and it creates a parabolic graph.
• A cubic function, such as \(f(x) = x^3\), has a degree of 3 and often displays an "S" shape curve.
• When these functions are composed with others, their degree can increase, impacting the shape and direction of their graphs.

Polynomial composition results, like \((f \circ g)(x) = f(g(x))\), can result in expressions such as \((x + 5)^2\) or \((x^3 + 3)\). These compositions are critical in determining the combined behavior of these functions. Understanding how polynomial functions transform through composition is essential, particularly for graphing.

Square Roots

The square root function is an essential concept in algebra. It represents a number which, when multiplied by itself, gives the original number under the root. The typical form is \( \sqrt{x} \), which has these important features:

• The domain is limited to non-negative numbers because square roots of negative numbers are not real.
• The shape is a curve that starts at the origin (0,0) and rises slowly, never decreasing.

When composed, such as \(f(g(x)) = \sqrt{x^2+4}\), the range begins at the square root of the smallest value inside the radical, giving an always non-negative result. These compositions must respect the domain constraints, potentially narrowing the input values you can use for graphing.

Domain and Range

Domain and range are essential in understanding functions' behavior:

• The domain is the set of all acceptable input values \(x\).
• The range includes all possible output values \(y\) after applying the function.

Function composition can affect both. For example, \((f \circ g)(x)\) could limit the domain compared to individual domains of \(f(x)\) or \(g(x)\). A square root example like \(f(g(x)) = \sqrt{x^2 - 4}\) necessitates \(x^2 - 4 \geq 0\), narrowing the domain to \(x \geq 2\) or \(x \leq -2\). Always examine composition constraints on domains or ranges before graphing.

Graphing Functions

Visualizing a function through graphing is a powerful tool for comprehension. Here's how to graph composed functions effectively:

• Plot the individual functions first, observing their shapes and intercepts.
• Next, compute the composition \((f \circ g)(x)\) and illustrate it on the same axes.
• Ensure to adjust for domain restrictions, which may affect where the graph is defined.

When graphing polynomial functions like \((g \circ f)(x) = g(f(x))\), consider transformations inherent in each function. For instance, the translation by adding \(c\) and stretching/shrinking when multiplicative terms appear alter the final graph's appearance. Accurate domain and range awareness will aid in plotting these complex pathways seamlessly.