Problem 28 - 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. $$(h-f)(0)$$

Step-by-Step Solution

Answer

The evaluated value of \((h-f)(0)\) is 1.

Step 1: Evaluate function f at x=0

First, substitute \(x = 0\) into \(f(x)\), which gives: \(f(0) = -(0)^2 + 0 = 0.\)

Step 2: Evaluate function h at x=0

Now, substitute \(x = 0\) into \(h(x)\), which gives: \(h(0) = -2(0) + 1 = 1.\)

Step 3: Subtract functions

Subtract the value of \(f(0)\) from \(h(0)\) as per the requirement of calculating \((h - f)(0)\). This gives: \((h - f)(0) = h(0) - f(0) = 1 - 0 = 1.\)

Key Concepts

Composite Functions

When we talk about composite functions in precalculus, we are referring to the combination of two functions in a specific order to create a new function. Essentially, the output of one function becomes the input for the second function. This is denoted as \(f(g(x))\) or \(g(f(x))\), depending on the order of composition.

It's crucial to follow the correct sequence here: evaluate the inner function first and then plug that result into the outer function. Imagine it as a nested process, where you start from the innermost part and work your way out. Failure to do so can lead to incorrect results, as order is key in function composition. To best understand how composite functions work, consider them as a 'function within a function', treating the combination as a single operation.

Function Evaluation

Function evaluation is essentially the act of finding the output of a function given a certain input. For the function \(f(x)\), when we say \(f(0)\), we're asking 'what does \(f\) output when \(x\) is zero?'. To evaluate a function, simply substitute the input value into the function and perform the necessary calculations.

As seen in the original exercise, when \(f(x)=-x^2+x\), evaluating \(f(0)\) means replacing \(x\) with 0, leading to the answer of 0. It is important to follow proper order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction), to ensure accuracy in function evaluation.

Function Subtraction

Function subtraction is one of the operations you can perform on functions, akin to addition, multiplication, and division of functions. The expression \( (h-f)(x) \) indicates that for every input \( x \), we should compute the output of \( h(x) \) and \( f(x) \) separately, and then subtract the output of \( f(x) \) from the output of \( h(x) \).

The step-by-step solution from the exercise exemplifies this process: after evaluating \( h(0) \) and \( f(0) \) independently, the difference is calculated by subtracting \( f(0) \) from \( h(0) \) to get the final answer. This is a fundamental concept in operations with functions and is often used in algebraic manipulations within precalculus.