Problem 24 - College Algebra

Question

Find: a. \((f \circ g)(x)\) b. \(\left(g^{\circ} f\right)(x)\) c. \((f \circ g)(2)\) $$f(x)=x^{2}+1, g(x)=x^{2}-3$$

Step-by-Step Solution

Answer

a. \((f \circ g)(x) = x^4 - 6x^2 + 10\), b. \((g \circ f)(x) = x^4 + 2x^2 - 2\), c. \((f \circ g)(2) = 2\).

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

To find \((f \circ g)(x)\), substitute \(g(x)\) into the function \(f(x)\). So, instead of \(x\) in \(f\), put \(g(x) = x^2 - 3\). Hence, \((f \circ g)(x) = (x^2 - 3)^2 + 1 = x^4 - 6x^2 + 10\).

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

To find \((g \circ f)(x)\), substitute \(f(x)\) into the function \(g(x)\). So, instead of \(x\) in \(g\), put \(f(x) = x^2 + 1\). Hence, \((g \circ f)(x) = (x^2 + 1)^2 - 3 = x^4 + 2x^2 - 2\).

Step 3: Find \((f \circ g)(2)\)

To find \((f \circ g)(2)\), substitute \(x = 2\) into the composite function \((f \circ g)(x)\). Hence, \((f \circ g)(2) = (2)^4 - 6*(2)^2 + 10 = 16 - 24 + 10 = 2\).

Key Concepts

Composite Functions

Composite functions are created when one function is applied to the result of another function. Think of it like a two-step process where one function feeds into another.
These composite functions are denoted as

• \((f \circ g)(x)\), which means applying function \(g\) first, then function \(f\) to that result,
• \((g \circ f)(x)\), which means applying function \(f\) first, then function \(g\) to that resulting value.

For example, if you have two functions, \(f(x) = x^2 + 1\) and \(g(x) = x^2 - 3\), the composite function \((f \circ g)(x)\) involves substituting \(g(x)\) into \(f(x)\), resulting in another polynomial expression.
By following this substitution process, you ensure the ordered application of functions maintains its integrity, allowing for complex transformations of input values.

Algebraic Functions

Algebraic functions are any functions composed using basic arithmetic operations such as addition, subtraction, multiplication, division, and root extraction with polynomials. These are some of the simplest forms of functions that can be written in an algebraic form.
The process of combining functions as seen in composite functions showcases algebraic manipulations. When we handle

• the substitution process,
• simplifying expressions like
  • taking powers of binomials, and
  • combining like terms,

we're fully engaged with algebraic function manipulation.
This type of function is beneficial for expressing more complex relationships using only basic algebra.

Polynomial Functions

Polynomial functions are a specific type of algebraic function that consists of variables raised to whole-number exponents and their coefficients. They look like \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each \(a\) is a constant.
In the example from the original problem, both \(f(x) = x^2 + 1\) and \(g(x) = x^2 - 3\) are polynomial functions.

• When composed together into the form \((f \circ g)(x) = (x^2 - 3)^2 + 1\), we again create another polynomial.
• This new polynomial shows how polynomials can be added, multiplied, or composed to create new polynomial functions.

Working with polynomial functions through composition involves

• evaluating expressions,
• using distributive property for expansion, and
• combining like terms,

which are all valuable skills in algebra.