Question

Find \((f+g)(x)\) and \((f-g)(x)\) and state the domain of each. Then evaluate \(f+g\) and \(f-g\) for the given value of \(x\). $$ f(x)=11 x+2 x^2, g(x)=-7 x-3 x^2+4 ; x=2 $$

Short answer

\((f+g)(x)=4x-x^2+4\), \((f-g)(x)=18x+5x^2-4\), \((f+g)(2)=8\), \((f-g)(2)=52\), and both functions have the domain of all real numbers.

Step-by-step solution

Find (f+g)(x)

Let's start by adding the two functions together. This is done by adding the coefficients of like terms. So, \(f(x) + g(x)=(11x+2x^2)+(-7x-3x^2+4)=(11x-7x)+(2x^2-3x^2)+4=4x-x^2+4\)

Find (f-g)(x)

Subtract \(g(x)\) from \(f(x)\). This is done by subtracting the coefficients of like terms. So, \(f(x) - g(x)=(11x+2x^2)-(-7x-3x^2+4)=(11x+7x)+(2x^2+3x^2)-4=18x+5x^2-4\)

Evaluate (f+g)(2)

Plug \(x=2\) into the equation for \(f+g\), resulting in \(4(2)-(2)^2+4=8-4+4=8\)

Evaluate (f-g)(2)

Now, plug \(x=2\) into the equation for \(f-g\), resulting in \(18(2)+5(2)^2-4=36+20-4=52\)

State the domain for each function

Since both \(f(x)+g(x)\) and \(f(x)-g(x)\) are polynomial functions, their domain is all real numbers.

Key concepts

Understanding Functions

A function is like a unique machine: for every input, it gives exactly one output. This is a crucial concept in mathematics. Functions are often represented by equations connecting input values to outputs.
For example, in our exercise, we have two functions:

• \(f(x) = 11x + 2x^2\)
• \(g(x) = -7x - 3x^2 + 4\)

These are polynomial functions, a specific type of function that includes powers of the variable \(x\). By working with functions, we can apply operations, like addition and subtraction, which helps us create new functions.

Exploring the Domain of a Function

The domain of a function refers to all possible input values (often \(x\)) that we can use without breaking any mathematical rules. For most polynomial functions, the domain is pretty straightforward: all real numbers.
In simpler terms, you can pick any number for \(x\) and the function will still work fine. In our example, both \((f+g)(x)\) and \((f-g)(x)\) are polynomials. Thus:

• The domain is all real numbers for both \((f + g)(x) = 4x - x^2 + 4\) and \((f - g)(x) = 18x + 5x^2 - 4\).

Domains become trickier with functions that involve fractions or square roots. Fortunately, that's not the case here!

Grasping Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers. These functions can be simple, like \(3x + 2\), or complex, like our examples:

• \(f(x) = 11x + 2x^2\)
• \(g(x) = -7x - 3x^2 + 4\)

They are defined by their terms, each being a product of a coefficient and a power of a variable. As seen here, polynomial functions can be combined through addition or subtraction. When added, like terms (terms with the same power) are combined by summing their coefficients.
This operation results in a new polynomial with its own expression as shown:

• Addition: \((f+g)(x) = 4x - x^2 + 4\)
• Subtraction: \((f-g)(x) = 18x + 5x^2 - 4\)

Understanding these basics will make it easier to manipulate and evaluate polynomial functions.