Problem 41 - Discrete Mathematics with Applications

Question

Using the functions \(f(x)=2 x+3\) and \(g(x)=x^{2}-1,\) find the following. $$(f+g)(-3)$$

Step-by-Step Solution

Answer

\( (f+g)(-3) = 5 \)

Step 1: Add the two functions f(x) and g(x)

To find the sum (f+g)(x), we will add the functions f(x) and g(x) together: \[ (f+g)(x) = f(x) + g(x) \] Using the given functions, plug in their respective equations: \[(f+g)(x) = (2x + 3) + (x^2 - 1)\]

Step 2: Simplify the sum of the functions

Combine the like terms and simplify the expression for (f+g)(x): \[(f+g)(x) = x^2 + 2x + 3 - 1\] \[(f+g)(x) = x^2 + 2x + 2\]

Step 3: Evaluate the sum of the functions at x = -3

Now that we have the simplified expression for (f+g)(x), we can substitute x = -3 to find the final answer: \[(f+g)(-3) = (-3)^2 + 2(-3) + 2\]

Step 4: Calculate the final answer

Perform the arithmetic operations within the expression: \[(f+g)(-3) = 9 - 6 + 2\] \[(f+g)(-3) = 3 + 2\] \[(f+g)(-3) = 5\] So, \((f+g)(-3) = 5\).

Key Concepts

Addition of Functions

When you're faced with two functions, such as \( f(x) = 2x + 3 \) and \( g(x) = x^2 - 1 \), often you'll want to combine them into a single function. This is accomplished through function addition. By adding two functions, you essentially add their outputs for the same input value.

Here's how you do it:

• First, write down the expressions for both functions.
• Then, simply add the corresponding expressions together.

For our example, this looks like:

• First, identify each function: \( f(x) = 2x + 3 \) and \( g(x) = x^2 - 1\).
• Add them to form a new function: \( (f+g)(x) = (2x + 3) + (x^2 - 1) \).

This results in a new function, \( (f+g)(x) = x^2 + 2x + 2 \), whose form can now be used for further calculations.

Evaluating Functions

Once you have a function like \( (f+g)(x) = x^2 + 2x + 2 \), the next step is often to find the function's value at a specific input, or to evaluate the function.

Evaluation simply means plugging a given value into the function in place of \( x \). This calculation gives you the output or the function's value for that particular input.

• Substitute the chosen value of \( x \) into the function.
• Calculate step-by-step to simplify the arithmetic operations.

For \( (f+g)(-3) \):

• Substitute \( x = -3 \) into the function: \( (-3)^2 + 2(-3) + 2 \).
• Calculate: \( 9 - 6 + 2 \).
• Simplify: \( 3 + 2 \).
• Finally, find: \( (f+g)(-3) = 5 \).

This tells us that when \( x = -3 \), the function \( (f+g)(x) \) results in the value 5.

Polynomial Simplification

In mathematics, combining like terms is something you'll encounter frequently—this process is part of simplifying polynomials. When you create a new function from two functions, like in our case with \( (f+g)(x) \), polynomial simplification is necessary.

Simplifying involves combining similar terms to make the expression cleaner and easier to work with.

Here's how you go about it:

• Look at each term in the function's sum: \( x^2 + 2x + 3 - 1 \).
• Combine constants: \( +3 - 1 = 2 \).
• Rewrite the function in its simplest form: \( x^2 + 2x + 2 \).

The result is a simplified polynomial, which makes it more straightforward when you're evaluating the function later. Keeping expressions simplified reduces errors and makes calculations faster, especially when multiple steps or complex expressions are involved.