Problem 38 - 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. $$\left(\frac{g}{h}\right)(-2)$$

Step-by-Step Solution

Answer

The result of evaluating \(\frac{g}{h}(-2)\) is -0.4.

Step 1: Understand Function Composition

A composite function \(\frac{g}{h}(x)\) means \(\frac{g(x)}{h(x)}\). This function is evaluated by first finding the output of function \(g\) at \(x\), the output of function \(h\) at \(x\) and then dividing the two results.

Step 2: Evaluate Function g

Substitute \(x = -2\) into \(g(x) = \frac{2}{x + 1}\) to get \(g(-2) = \frac{2}{-2 + 1} = -2\).

Step 3: Evaluate Function h

Similarly, substitute \(x = -2\) into \(h(x) = -2x + 1\) to get \(h(-2) = -2(-2) + 1 = 5\).

Step 4: Evaluate the Composite Function

Now we can evaluate the composite function by dividing the output of \(g(-2)\) by \(h(-2)\). This gives us \(\frac{g}{h}(-2) = \frac{-2}{5} = -0.4\).

Key Concepts

Composite Functions

Composite functions involve combining two or more functions to create a new function. A common way to denote a composite function is using a fraction, such as \( \frac{g}{h}(x) \). This means you should first find the values of \(g(x)\) and \(h(x)\). After you get these results, you perform the arithmetic operation indicated, in this case, division.

Understanding how composite functions work is crucial for solving problems where multiple functions interact. It's important to methodically evaluate each function separately before combining them as per the required operation.

• Step one is to evaluate the functions individually using the given value of \(x\).
• Step two is to follow through with the arithmetic operation once the individual function values are known.

This step-by-step approach helps in simplifying complex operations and reduces the possibility of errors.

Evaluating Functions

Evaluating functions means finding the output of a function for a given input value of \(x\). It involves substituting a specific value into the function equation. The right substitution and computation will yield the function's result for that input value.

For example, if you have \(g(x) = \frac{2}{x+1}\) and want to evaluate it at \(x = -2\):

• Substitute \(-2\) in place of \(x\) in the function: \(g(-2) = \frac{2}{-2 + 1} \).
• Simplify the expression to get the result: \(g(-2) = -2\).

When completing this process, careful calculation is necessary to ensure accurate results. Evaluating is a fundamental skill and helps in dealing with more complex function problems confidently.

Rational Expressions

Rational expressions involve both polynomials in the numerator and the denominator. They form fractions, much like the composite function \( \frac{g}{h}(x) \), where each component or function is rational. These expressions can be simple, like the one in this exercise, or more complex with higher degree polynomials.

When dealing with rational expressions:

• Evaluate the function separately for the numerator and the denominator.
• Ensure that the denominator is not zero, as division by zero is undefined in mathematics.

Once you find the values of both the numerator and denominator, you can finish the division. In our example, the division was simple:

• \(\frac{g(-2)}{h(-2)} = \frac{-2}{5}\).
• This leads to a result of \(-0.4\).

Understanding and working with rational expressions is an invaluable skill, as it paves the way for tackling more complex algebraic operations.