Problem 7 - A Graphical Approach to College Algebra

Question

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$(f \circ g)(3)$$

Step-by-Step Solution

Answer

108

Step 1: Understand the Function Composition

The composition of two functions, denoted as \((f \circ g)(x)\), means \(f(g(x))\). This indicates that we need to substitute \(g(x)\) into the function \(f(x)\).

Step 2: Substitute g(x) into f(x)

Given \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\), substitute \(g(x)\) into \(f(x)\): \[ f(g(x)) = ((2x - 1)^2 + 3(2x - 1)) \]

Step 3: Simplify the Expression

Expand \((2x - 1)^2\): \((2x - 1)^2 = 4x^2 - 4x + 1\). Now substitute this result back into \(f(g(x))\): \[ f(g(x)) = 4x^2 - 4x + 1 + 6x - 3 \] Simplify to: \[ f(g(x)) = 4x^2 + 2x - 2 \]

Step 4: Evaluate f(g(x)) at x = 3

Now we need to evaluate \(f(g(3))\). First calculate \(g(3) = 2(3) - 1 = 5\). Substitute \(x = 5\) into \(f(x)\): \[ f(5) = 4(5)^2 + 2(5) - 2 \] Calculate further: \[ f(5) = 4(25) + 10 - 2 = 108 \]

Key Concepts

Polynomial Function Basics

A polynomial function is a mathematical expression that consists of variables raised to whole number powers and coefficients. The general form of a polynomial is:

• The function has the form: \(a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\), where \(n\) is a non-negative integer.
• Each term like \(a_nx^n\) has a coefficient \(a_n\) and a variable \(x\) raised to an exponent \(n\).
• The highest power of a variable, \(n\), describes the degree of the polynomial.

To explain with the example given, \(f(x) = x^2 + 3x\) is a polynomial.
It has two terms: \(x^2\) (with a power of 2) and \(3x\), each having coefficients of 1 and 3, respectively.
The highest power here is 2, making this a quadratic polynomial, a specific type of polynomial.

Understanding Function Evaluation

Evaluating a function means finding the value of the function for a specific input.
This involves substituting the input value into the function equation and simplifying it to find the result.
In our example, let's understand how we evaluate functions step-by-step:

• Given function \(g(x) = 2x - 1\), evaluate it for \(x = 3\). Substitute 3 in place of \(x\) in the function equation.
• This becomes \(g(3) = 2(3) - 1 = 6 - 1 = 5\).

Next, we use this result to evaluate \(f(g(3))\) or \(f(5)\) using \(f(x) = x^2 + 3x\).
Replacing \(x\) with 5:

• Calculate \(f(5) = (5)^2 + 3(5) = 25 + 15 = 40\).

Evaluating functions is crucial for applying mathematical operations or modeling situations.

Steps to Simplify Expressions

Simplifying expressions refers to reducing them to their simplest form.
This often involves combining like terms, applying arithmetic operations, and ensuring that the expression is as compact as possible.
Let's see how simplification is applied in the function composition:

• After substituting \(g(x) = 2x - 1\) into \(f(x) = x^2 + 3x\), we have: \[f(g(x)) = (2x - 1)^2 + 3(2x - 1)\].
• Expand \((2x - 1)^2\) which is \(4x^2 - 4x + 1\).
• Now distribute and add: \[4x^2 - 4x + 1 + 6x - 3\]This combines to: \[4x^2 + 2x - 2\]

These steps help clarify how operations within expressions are handled and combined into something simpler and more understandable.