Problem 81 - College Algebra with Corequisite Support

Question

For the following exercises, use \(f(x)=x^{3}+1\) and \(g(x)=\sqrt[3]{x-1}\). Find \((f \circ g)(2)\) and \((g \circ f)(2)\)

Step-by-Step Solution

Answer

Both compositions, \((f \circ g)(2)\) and \((g \circ f)(2)\), equal 2.

Step 1: Understanding Function Composition

Function composition involves substituting one function into another. If you have two functions, \( f(x) \) and \( g(x) \), then \( (f \circ g)(x) \) means you apply \( g(x) \) first, and then use the result as the input for \( f(x) \). Similarly, \( (g \circ f)(x) \) involves applying \( f(x) \) first and then using the result as the input for \( g(x) \).

Step 2: Calculate \((f \, \circ \, g)(2)\)

First, find \(g(2)\):\[ g(x) = \sqrt[3]{x-1} \]Plug in \( x = 2 \):\[ g(2) = \sqrt[3]{2-1} = \sqrt[3]{1} = 1 \]Now, use the result of \( g(2) \) to find \( f(g(2)) \):\[ f(x) = x^3 + 1 \]Plug in \( g(2) = 1 \):\[ f(1) = 1^3 + 1 = 2 \]

Step 3: Calculate \((g \, \circ \, f)(2)\)

First, find \(f(2)\):\[ f(x) = x^3 + 1 \]Plug in \( x = 2 \):\[ f(2) = 2^3 + 1 = 8 + 1 = 9 \]Now, use the result of \( f(2) \) to find \( g(f(2)) \):\[ g(x) = \sqrt[3]{x-1} \]Plug in \( f(2) = 9 \):\[ g(9) = \sqrt[3]{9-1} = \sqrt[3]{8} = 2 \]

Step 4: Review Results

Therefore, after computing each composition separately, we have: \((f \circ g)(2) = 2\) and \((g \circ f)(2) = 2\).

Key Concepts

Understanding Cubic Functions

Cubic functions are polynomials of degree three, meaning they have an equation of the form \(ax^3 + bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(aeq0\). In our exercise, the cubic function given is \(f(x) = x^3 + 1\). This specific function simplifies to have coefficients: \(a=1\), \(b=0\), \(c=0\), and \(d=1\). The cubic term \(x^3\) is crucial because it dominates the behavior of the function, especially for larger values of \(x\). The positive coefficient of \(x^3\) ensures the function increases to infinity as \(x\) becomes more positive or more negative. The constant term, in this case, '+1', acts to shift the entire curve upwards on the graph.Some key characteristics of cubic functions include:

• They have at least one real root, since the graph of the function goes to infinity as \(x\) approaches positive or negative infinity.
• Their graphs typically have a change in direction, giving a characteristic 'S' shape.
• These functions exhibit local maximums or minimums, which occur where the derivative is zero.

Exploring Cube Root Functions

Cube root functions are the inverse of cubic functions. These functions take a number and find the value which, when cubed, gives the original number as the result. The notation for a cube root function is \(g(x) = \sqrt[3]{x}\). The specific function in this exercise is \(g(x) = \sqrt[3]{x-1}\). In this case, the function involves a horizontal shift by one unit to the right due to the '-1' inside the root.Key features of cube root functions include:

• Being defined for all real numbers, unlike square roots which are undefined for negative numbers.
• Having one real root, meaning they cross the x-axis at one point, which is also the point where their corresponding cubic function equals zero.
• Behaving symmetrically with respect to the origin, if neither shifted nor manipulated otherwise.
• Exhibiting continuous growth, i.e., continually increasing but at a decreasing rate as \(x\) progresses.

In solving equations, the purpose of the cube root is to eliminate the cube, making cube rooting an essential tool for unraveling cubic equations.

Significance of Mathematical Operations and Function Composition

Mathematical operations in function composition allow one to form new functions by combining existing ones. This capability is significant because it offers more complex structures and behaviors from simpler foundational functions. Understanding how to manipulate these operations is crucial for solving composite functions like \((f \circ g)(x)\) and \((g \circ f)(x)\) in exercises. Function composition involves:

• Reversing the usual order of operations; you apply the inner function first and then the outer function on the result.
• Generating functions that can be more intricate in behavior and produce interesting outputs.
• Needing careful substitution, ensuring every step in the composition is followed exactly to avoid errors.

In our solution, applying \(g(x) = \sqrt[3]{x-1}\) first in \((f \circ g)(2)\), resulted in the transformation \((x-1)\) becoming 1, which \(f(x) = x^3 + 1\) then cube to revert to 2, exemplifies how function operations knit together to build and solve problems.