Problem 37 - Precalculus : Building Concepts and Connections

Question

In Exercises \(17-40,\) let \(f(x)=-x^{2}+x, g(x)=\frac{2}{x+1},\) and \(h(x)=-2 x+1 .\) Evaluate each of the following. $$\left(\frac{g}{h}\right)(3)$$

Step-by-Step Solution

Answer

The value of the function \(\left(\frac{g}{h}\right)(3)\) is -0.1.

Step 1: Calculate the value of g(3)

Substitute \(x = 3\) into the function \(g(x)=\frac{2}{x+1}\) and solve for the value. Thus, \(g(3) = \frac{2}{3+1} = \frac{2}{4} = 0.5\).

Step 2: Calculate the value of h(3)

Substitute \(x = 3\) into the function \(h(x) = -2x + 1\) and solve for the value. Thus, \(h(3) = -2(3) + 1 = -6 + 1 = -5\).

Step 3: Find the value of \(\left(\frac{g}{h}\right)(3)\)

Now substitute \(g(3)\) and \(h(3)\) into \(\left(\frac{g}{h}\right)(x)\) and evaluate. Therefore, \(\left(\frac{g}{h}\right)(3) = \frac{g(3)}{h(3)} = \frac{0.5}{-5} = -0.1\).

Key Concepts

Function Evaluation

Understanding the evaluation of functions is crucial for solving problems involving rational functions like the one presented in the exercise. Function evaluation simply means finding the output of a function for a specific input.
To evaluate a function, substitute the given input value into the function's equation and perform the necessary calculations. Let's see how this works with the function evaluation process using functions that include polynomials or rational expressions.

• Take the function you need to evaluate, for example, \( g(x) = \frac{2}{x+1} \).
• Substitute the specific value, \( x = 3 \), into the equation.
• Perform the arithmetic operations: \( g(3) = \frac{2}{3+1} = 0.5 \).

Function evaluation provides the "height" of the function for specific inputs and is necessary when calculating combined functions like \( \left(\frac{g}{h}\right)(3) \) in this scenario.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. The provided exercise deals with finding the value of a rational expression, which involves both polynomial and rational expressions.
In the specific rational function \( \left(\frac{g}{h}\right)(x) \), the process of evaluating involves determining the value of both \( g(x) \) and \( h(x) \) separately before combining them:

• Identify and evaluate each expression individually, such as \( g(3) = 0.5 \) and \( h(3) = -5 \).
• Use the evaluated values to compute the rational expression \( \left(\frac{g}{h}\right)(3) = \frac{0.5}{-5} = -0.1 \).

This process underscores the importance of precision in arithmetic to simplify and solve such algebraic expressions accurately.

Composite Functions

Composite functions involve the combination of two or more functions, often expressed in the form \( (f \circ g)(x) = f(g(x)) \). However, in our exercise, the combination takes the form of division, \( \left(\frac{g}{h}\right)(x) \).
Understanding how to evaluate composite functions or combined functions involves multiple steps:

• Evaluate each individual function for the chosen input, as seen with \( g(x) \) and \( h(x) \).
• Combine the resulting values appropriately to evaluate the overall expression.
• In this context, you deal with the division of results from separate evaluations, thus incorporating both function operations and arithmetic.

This methodical approach lets you understand the interrelation between functions, enhancing your ability to solve complex equations by breaking them down into simpler parts.