Problem 24 - 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. $$(f-g)(-3)$$

Step-by-Step Solution

Answer

The value of (f-g)(-3) is -11.

Step 1: Define the functions and the value of x

First, we define the functions and the value of x which we will use.\nGiven that \(f(x)=-x^{2}+x\), \(g(x)=\frac{2}{x+1}\), and we're calculating \(f-g\) at x = -3.

Step 2: Compute f(x) and g(x)

We compute the function \(f(-3)\) and function \(g(-3)\). For \(f(-3)\), replace x in \(f(x) = -x^{2}+x\) with -3 to get \(-(-3)^{2}+(-3) = -9+(-3)=-12\). For \(g(-3)\), replace x in \(g(x) = \frac{2}{x+1}\) with -3 to get \(\frac{2}{-3+1}=\frac{2}{-2}=-1\).

Step 3: Subtract g from f at x = -3

We now compute the function \(f(-3)-g(-3)\) by subtracting the previously calculated values. So, \(-12-(-1) = -12+1 = -11.\)

Key Concepts

Function Operations

Understanding function operations is foundational in precalculus, as it enables students to manipulate and combine functions in various ways.

One basic operation is the 'difference of functions.' It involves taking two functions, say, \(f(x)\) and \(g(x)\), and constructing a new function by subtracting the second function from the first. Symbolically, this is denoted as \((f-g)(x) = f(x) - g(x)\).

To perform the operation, you simply evaluate both functions separately at the same value of \(x\) and then subtract the result of \(g(x)\) from that of \(f(x)\). This new function, \((f-g)(x)\), inherits the domain from both \(f(x)\) and \(g(x)\), but with the consideration that it cannot include points where \(g(x)\) would cause the expression to be undefined.

Substituting Values in Functions

When you have functions defined by certain algebraic expressions, substituting values in these functions is a process of evaluating their behavior at specific points. It's a key skill in precalculus and is crucial for understanding how functions operate.

To substitute a value into a function, you replace the variable, often \(x\), with a numerical value. For example, if you have the function \(f(x) = -x^2 + x\) and you want to find \(f(-3)\), you replace every instance of \(x\) in the function's equation with -3 and then simplify. Here’s how it’s done:

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

When getting acquainted with functions, practicing this substitution method can fundamentally improve your understanding and fluency when working with more complicated calculus concepts.

Difference of Functions

The 'difference of functions' occurs when you have the subtraction of one function from another. In our example, the difference of function \(f\) and function \(g\) at \(x = -3\) involves two main steps: evaluating each function separately at \(x = -3\) and then subtracting the value of \(g(-3)\) from \(f(-3)\).

The subtraction is quite straightforward once you have the values from each function. Here is how you calculate it:

• First, evaluate \(f(-3)\) which gives \(-12\).
• Next, evaluate \(g(-3)\) resulting in \(-1\).
• Finally, subtract to get \(f(-3) - g(-3) = -12 - (-1) = -11\).

This operation is a fundamental aspect of working with functions and sets the basis for more intricate calculus topics such as composing functions and finding limits. It also provides insight into how functions can be manipulated graphically; in this case, by shifting the graph of one function 'down' by the value of another.