Problem 27 - Precalculus with Limits

Question

In Exercises 17-28, evaluate the indicated function for \(f(x) = x^2 + 1\) and \(g(x) = x - 4\). \((f/g)(-1)-g(3)\)

Step-by-Step Solution

Answer

The solution to \((f/g)(-1)-g(3)\) for the given functions \(f(x)\) and \(g(x)\) is 0.6.

Step 1: Identify the function \((f/g)(x)\)

The function \((f/g)(x)\) means that for each value of \(x\), \(f(x)\) will be divided by \(g(x)\). In this case, this means \((x^2 + 1)/(x - 4)\).

Step 2: Substitute \(x = -1\) in \((f/g)(x)\)

Substitute the value of \(x\) in the expression we found in Step 1. Therefore, \((f/g)(-1)=((-1)^2 + 1)/((-1) - 4) = 2/(-5) = -0.4.\)

Step 3: Evaluate \(g(3)\)

Substitute the value of \(3\) in the function \(g(x)\). Therefore, \(g(3) = 3 - 4 = -1.\)

Step 4: Combine the results

Now that we have evaluated both \((f/g)(-1)\) and \(g(3)\), we can plug these values into the original expression, \((f/g)(-1)-g(3) = -0.4 - (-1) = 0.6.

Key Concepts

Algebraic Functions

Algebraic functions form a fundamental concept in mathematics. These functions involve variables and constants combined with arithmetic operations like addition, subtraction, multiplication, and division. In the context of the exercise, we have two such algebraic functions, \(f(x) = x^2 + 1\) and \(g(x) = x - 4\). Each of these defines an operation on variable \(x\):

• \(f(x)\) involves squaring the variable \(x\) and adding 1. It is a quadratic function.
• \(g(x)\) involves subtracting 4 from \(x\), making it a linear function.

These functions help us understand relationships between numbers, and they can be manipulated using different operations to solve various math problems. Algebraic functions are the backbone of equations and are critical for interpreting how changes in one quantity can affect another.

Function Operations

Function operations, as shown in the exercise, allow us to perform arithmetic operations with functions. This involves combining functions to form new ones such as addition \((f + g)(x)\), subtraction \((f - g)(x)\), multiplication \((f \cdot g)(x)\), and division \((f/g)(x)\). In our example, we specifically explored how the division of two functions works:

• To perform \((f/g)(x)\), we divide the function \(f(x)\) by \(g(x)\).
• As demonstrated, it is crucial to evaluate each function algebraically before performing the division to avoid errors.

This type of operation is widely used in mathematical modeling and problem solving, allowing us to explore how different variables interact with each other. By mastering function operations, students can broaden their problem-solving toolkit and approach more complex mathematical scenarios with confidence.

Substitution Method

The substitution method is a powerful technique used to evaluate functions by replacing variables with given numbers. In the solution steps, the substitution method was applied to find the values needed:

• First, substitute \(x = -1\) into \((f/g)(x)\) to get \((-1)^2 + 1)/((-1) - 4) = 2/(-5) = -0.4.\)
• Then, substitute \(x = 3\) into \(g(x)\) yielding \(3 - 4 = -1.\)

Using substitution can simplify complex functions by reducing them to numerical expressions that are easier to handle. This method requires careful calculation, ensuring that each substitution is correctly executed, so that the final result accurately reflects the intended computation. Practicing substitution enables students to become more adept at handling functional evaluations in various mathematical contexts.