Problem 27 - Beginning and Intermediate Algebra

Question

Let \(m(x)=x+8\) and \(n(x)=-x^{2}+3 x-8 .\) Find a) \((n \circ m)(x)\) b) \(\quad(m \circ n)(x)\) c) \(\quad(m \circ n)(0)\)

Step-by-Step Solution

Answer

a) \((n \circ m)(x) = -(x+8)^2+3(x+8)-8\) b) \((m \circ n)(x) = (-x^2 + 3x )\) c) \((m \circ n)(0) = 0\)

Step 1: Write down the given functions

The given functions are: \(m(x) = x+8\) and \(n(x) = -x^2 + 3x -8\).

Step 2: Find n(m(x))

Now we want to find the composition of n and m, which is written as \(n(m(x))\). First, we replace x in the definition of the function n(x) with the function m(x): \[n(m(x))= -(x+8)^2+3(x+8)-8\] b) Finding \((m \circ n)(x)\)

Step 3: Write down the given functions

The given functions are: \(m(x) = x+8\) and \(n(x) = -x^2 + 3x -8\).

Step 4: Find m(n(x))

Now we want to find the composition of m and n, which is written as \(m(n(x))\). First, we replace x in the definition of the function m(x) with the function n(x): \[m(n(x))= (-x^2 + 3x -8)+8\] c) Finding \((m \circ n)(0)\)

Step 5: Evaluate n(0) and m(n(0))

First, we need to find the value of n(0): \\ n(0) = - (0)^2 + 3(0) - 8\\ n(0) = -8\\ Now, we can find the value of m(n(0)): \\ m(n(0)) = m(-8)\\ m(-8) = -8 + 8 Now, let's write the final answers together: a) \((n \circ m)(x) = -(x+8)^2+3(x+8)-8\) b) \((m \circ n)(x) = (-x^2 + 3x -8)+8\) c) \((m \circ n)(0) = m(n(0)) = m(-8) = -8+8 = 0\)

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions that involve operations such as addition, subtraction, multiplication, division, and powers with algebraic expressions. In the context of function composition, these functions are crucial because they form the basis of variables and operations that will be combined.

• In algebraic functions like the ones given: \(m(x) = x + 8\) and \(n(x) = -x^2 + 3x - 8\), each operation is fundamental for forming the function's overall expression.
• When working with algebraic functions, it's essential to understand that operations follow specific rules such as precedence, which dictates the order in which operations are performed.

A composed function, such as \(n(m(x))\), utilizes these fundamental operations, as it involves substituting the value of \(m(x)\) into \(n(x)\).
Understanding algebraic functions and their compositions helps us transform complex equations into understandable and solvable expressions.

Evaluating Functions

Evaluating functions means finding the output of a function for a specific input. When evaluating individual functions, we substitute the given input value into the expression.

• For example, if \(x = 0\), to evaluate \(n(x)\), you substitute 0 into \(n(x) = -x^2 + 3x - 8\) and solve it. This simplifies to \(n(0) = -8\).
• It's important to maintain the correct order of operations when evaluating each part of the function. This ensures precise calculations.

Evaluating composed functions, such as \((m \circ n)(x)\), follows a similar procedure. First, evaluate the innermost function which, in this case, is \(n(x)\), then substitute its output into \(m(x)\).
When dealing with function compositions, it's also vital to ensure that you're substituting correctly by double-checking each step, especially with compounded expressions.

Polynomial Functions

Polynomial functions are a subtype of algebraic functions characterized by expressions involving powers of variables added together. They are represented as \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where \(a_n\) are coefficients.

• The function \(n(x) = -x^2 + 3x - 8\) is an example of a polynomial function of degree 2, known as a quadratic polynomial.
• These functions are particularly important because they form the building blocks for more complex compositions.

Polynomial functions exhibit properties such as continuity and smoothness, which makes them incredibly powerful for modeling real-world scenarios.
In the process of function composition, knowing the behavior of polynomial functions, like their roots and curvature, aids in predicting the nature of the resulting composition.