Problem 15 - 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. $$\left(\frac{f}{g}\right)(-1)$$

Step-by-Step Solution

Answer

\(\left(\frac{f}{g}\right)(-1) = \frac{2}{3}\)

Step 1: Understand the Composition

The function \(\left(\frac{f}{g}\right)(x)\) represents the division of the function \(f(x)\) by \(g(x)\). Mathematically, it is expressed as \(\frac{f(x)}{g(x)}\). The task is to evaluate this expression at \(x = -1\).

Step 2: Calculate \(f(-1)\)

To find \(f(-1)\), substitute \(-1\) into the function \(f(x) = x^2 + 3x\): \[f(-1) = (-1)^2 + 3(-1) = 1 - 3 = -2\]

Step 3: Calculate \(g(-1)\)

To find \(g(-1)\), substitute \(-1\) into the function \(g(x) = 2x - 1\): \[g(-1) = 2(-1) - 1 = -2 - 1 = -3\]

Step 4: Evaluate \(\frac{f(-1)}{g(-1)}\)

We now compute the division of the results from Step 2 and Step 3: \[\frac{f(-1)}{g(-1)} = \frac{-2}{-3} = \frac{2}{3}\]

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number exponents. They are composed of constants and coefficients which are also sometimes referred to as terms. A polynomial function is in the standard form of:

• For example, the function \(f(x) = x^2 + 3x\) is a polynomial function. It consists of two terms: \(x^2\) and \(3x\). The highest degree (or power of the variable) in this function is 2, making it a quadratic polynomial.

Polynomials are continuous, meaning there are no breaks or holes in their graphs. They can be added, subtracted, and multiplied, making them versatile for various types of algebraic operations.
To evaluate a polynomial function like \(f(x)\) at a specific point, such as \(x = -1\), substitute the value of the variable with the number and perform the arithmetic calculations. In the exercise, this manipulation resulted in \(f(-1) = -2\).

Rational Functions

Rational functions are defined as fractions of polynomial functions. The general form is a ratio of two polynomial functions, such as \(\frac{f(x)}{g(x)}\).
These functions can have asymptotes and discontinuities, depending on the zeroes of the polynomial in the denominator. For example, in the exercise solution, \(\left(\frac{f}{g}\right)(x)\) indicates rational function where \(f(x)\) and \(g(x)\) are polynomials.

• For instance, \(g(x) = 2x - 1\) is the denominator, which influences the function’s domain. It's critical to ensure that the denominator does not equate to zero, as this would make the function undefined at that point.

When calculating \(\left(\frac{f}{g}\right)(-1)\), you substitute \(-1\) into both the numerator and the denominator polynomials resulting in \(\frac{-2}{-3}\), simplifying to \(\frac{2}{3}\).

Function Evaluation

Function evaluation involves substituting the given value of \(x\) into the function and solving for the output. It’s a fundamental concept that helps in understanding how the function operates and locating specific points on a graph.
For the exercise, evaluating \(f(x)\) and \(g(x)\) separately was done by replacing \(x\) with \(-1\) in both functions. This step-by-step process ensures accuracy and clarity.

• Evaluating \(f(-1)\) gave \(-2\) and \(g(-1)\) provided \(-3\).
• The accurate evaluation of these values forms the basis of correctly computing the rational function \(\left(\frac{f}{g}\right)(-1)\).

Understanding this substitution method makes it simpler to tackle more complex compositions and operations in both polynomial and rational functions.