Problem 19 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The result is \(-3\).

Step 1: Understand the Operation

The expression \((g-f)(-2)\) represents the result of subtracting the function \(f(x)\) from the function \(g(x)\) before evaluating at \(x = -2\). We will calculate \((g(x) - f(x))\) first.

Step 2: Determine the Expression for \(g(x) - f(x)\)

To find \((g(x) - f(x))\), substitute the expressions for \(g(x)\) and \(f(x)\):\[g(x) = 2x - 1\]\[f(x) = x^2 + 3x\]Now, calculate \(g(x) - f(x)\):\[(g(x) - f(x)) = (2x - 1) - (x^2 + 3x)\]Simplifying the expression:\[-x^2 - x - 1\].

Step 3: Evaluate \((g-f)(-2)\)

Substitute \(x = -2\) into the expression \(-x^2 - x - 1\):\(-(-2)^2 - (-2) - 1\)Calculate each term:\[\text{First term: } -(-2)^2 = -4\]\[\text{Second term: } -(-2) = +2\]\[\text{Constant term: } -1\]Sum the values:\(-4 + 2 - 1 = -3\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions that involve sums of powers of variables with constant coefficients. A polynomial function is generally represented as:

• a0 + a1x + a2x^2 + ... + anx^n

Here, the coefficients \(a0, a1, \ldots , an\) are constants, and the powers \(0, 1, 2, \ldots , n\) are non-negative integers.
In our exercise, we are dealing with two polynomial functions, namely \(f(x)=x^2 + 3x\) and \(g(x)=2x - 1\).
Each term in a polynomial function involves a variable \(x\) raised to a power, and the coefficients are fixed values. The degree of a polynomial is defined as the highest power of the variable \(x\) present in the function. For example, the degree of \(f(x)\) is 2 because the highest power of \(x\) is 2.
Understanding polynomials is crucial as they serve as a foundation for dealing with more complex mathematical concepts like calculus.

Function Subtraction

Function subtraction involves creating a new function by subtracting one function from another. If you have two functions \(f(x)\) and \(g(x)\), the subtraction \(g(x) - f(x)\) results in another function that shows the difference between the two, for each value of \(x\).
To perform function subtraction, simply subtract the terms of \(f(x)\) from \(g(x)\) term by term. Using the exercise's functions, we subtract the terms from \(f(x) = x^2 + 3x\) from \(g(x) = 2x - 1\):

• Start by organizing the terms: \(2x - 1 - (x^2 + 3x)\)
• Distribute the negative sign through \(f(x)\) to ensure correct subtraction: \(-x^2 - 3x\)
• Combine like terms: \-x^2 - x - 1\

The subtraction of the functions yields another polynomial, \-x^2 - x - 1\, which we can use for further evaluations.
Function subtraction is helpful in applications where understanding the difference between two processes or datasets is essential.

Function Evaluation

Function evaluation involves finding the value of a function at a specific point, which means substituting a particular value for the variable and simplifying the expression to get a result.
When given a function like \(h(x) = -x^2 - x - 1\), evaluating \(h(-2)\) requires substituting \(x = -2\) into the function, and then simplifying:

• Substitute \(-2\) into the expression: \(-(-2)^2 - (-2) - 1\)
• Simplify each term:
• First term: \(-(-2)^2 = -4\)
• Second term: \(+2\)
• Constant term: \(-1\)

Total: \(-4 + 2 - 1 = -3\)
This process shows that \(h(-2) = -3\).
Function evaluation is a fundamental concept as it applies to many real-world problems such as calculating speeds, profit estimations, and more by providing actual numerical results based on mathematical models.