Problem 31 - 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. $$(g h)(-3)$$

Step-by-Step Solution

Answer

The result of the function composition \((gh)(-3)\) is 0.25.

Step 1: Evaluate h(-3)

Substitute -3 into the function \(h(x)\): \(h(-3) = -2(-3) + 1 = 6 + 1 = 7\).

Step 2: Substitute h(-3) into g

Now, use the result of h(-3) = 7 from step 1 and substitute into g(x): \(g(7) = \frac{2} {7 + 1} = \frac{2} {8} = 0.25\).

Key Concepts

Function Composition

Function composition is essentially a 'function of a function' where the output of one function becomes the input of another. It's comparable to a sequence of operations where you perform one task and use its result as the starting point for the next task. In mathematical terms, if you have two functions, say f and g, and you want to compose them (denote it as \(g \textcircled{ } f\)), this means you first apply f and then apply g to the result of f. For instance, if you were to compute \(g \textcircled{ } h)(-3)\), you would follow two steps:

1. Evaluate \(h(-3)\) - find out what h does to the input of -3.
2. Take the result from step 1 and use it as an input for g, leading to \(g(h(-3))\).

It's essential to follow the correct order when composing functions since \(f \textcircled{ } g\) can yield a different result compared to \(g \textcircled{ } f\). The concept of composition is fundamental in more advanced mathematics, including calculus, where it contributes to understanding the chain rule for derivatives.

Evaluating Functions

Evaluating functions is a critical skill in precalculus that involves substituting a specific value for the variable in a function's formula and simplifying to find the result. You can think of a function as a machine that takes an input, processes it according to a certain rule (the function expression), and then produces an output. For instance, if you have a function \(f(x) = -x^2 + x\), and you want to evaluate this function at \(x = -3\), you'd substitute -3 for every instance of \(x\) in the formula:

\(f(-3) = -(-3)^2 + (-3) = -9 - 3 = -12\).

It's important to pay close attention to signs and operations to avoid common mistakes, such as squaring negatives incorrectly or forgetting to distribute multiplication over addition or subtraction.

Function Operations

Function operations allow us to perform arithmetic on functions just as we would with numbers. This toolkit includes addition, subtraction, multiplication, division, and composition of functions. When combining functions through these operations, you apply the operation to the outputs of your functions for any input x.

For example:

• Addition: \( (f+g)(x) = f(x) + g(x) \)
• Subtraction: \( (f-g)(x) = f(x) - g(x)\)
• Multiplication: \( (f \cdot g)(x) = f(x) \cdot g(x)\)
• Division: \( (f \/ g)(x) = \frac{f(x)}{g(x)} \), given that \(g(x) \eq 0\)

These operations can be used to create new functions with properties that are helpful in various applications, such as physics and engineering. The exercise improvement would come from understanding these operations and how to apply them carefully to avoid mistakes, especially in the cases where the order of operations must be strictly followed to obtain the correct output.