Problem 84 - Elementary Algebra

Question

Calculate \((f+g)(x)\) and \((f-g)(x)\) and state the restrictions to the domain. \(f(x)=x x-5\) and \(g(x)=12 x-3\)

Step-by-Step Solution

Answer

\((f+g)(x) = x^2 + 12x - 8\), \((f-g)(x) = x^2 - 12x - 2\), domain: all real numbers.

Step 1: Add Functions

To find \((f+g)(x)\), add the given functions: \[ f(x) + g(x) = (x^2 - 5) + (12x - 3) \]Combine like terms:\[ f(x) + g(x) = x^2 + 12x - 5 - 3 \]Which simplifies to:\[ f(x) + g(x) = x^2 + 12x - 8 \]

Step 2: Subtract Functions

To find \((f-g)(x)\), subtract the function \(g(x)\) from \(f(x)\):\[ f(x) - g(x) = (x^2 - 5) - (12x - 3) \]Distribute the subtraction across the terms:\[ f(x) - g(x) = x^2 - 5 - 12x + 3 \]Combine like terms:\[ f(x) - g(x) = x^2 - 12x - 2 \]

Step 3: Determine Domain Restrictions

Identify any restrictions on the domain for both \(f(x)\) and \(g(x)\). Since both functions are polynomials (\(x^2 - 5\) and \(12x - 3\)), they are defined for all real numbers. Hence, the domain for \((f+g)(x)\) and \((f-g)(x)\) is also all real numbers.

Key Concepts

Function Addition

When we talk about function addition, we mean adding together the outputs of two functions for the same input. For this exercise, we are given two functions, \(f(x) = x^2 - 5\) and \(g(x) = 12x - 3\). The expression \((f+g)(x)\) denotes the sum of these functions.
To find this, simply add the functions term by term:

• Add the \(x^2\) term from \(f(x)\) to the \(12x\) term from \(g(x)\). Since they are not the same type of terms, they stay as they are.
• Combine the constant terms. In this case, \(-5\) from \(f(x)\) and \(-3\) from \(g(x)\) add up to \(-8\).

Putting it all together, we have:\[ (f+g)(x) = x^2 + 12x - 8 \]This new function combines all effects of \(f\) and \(g\) where each input is concerned, making addition of functions a straightforward technique.

Function Subtraction

Function subtraction, while similar to addition, requires us to subtract one function's output from another's. The expression \((f-g)(x)\) indicates the difference between two functions for the same input value. We are working with the functions \(f(x) = x^2 - 5\) and \(g(x) = 12x - 3\).
To find \((f-g)(x)\), subtract the terms of \(g(x)\) from \(f(x)\):

• Retain the \(x^2\) from \(f(x)\) as there is no corresponding \(x^2\) term in \(g(x)\).
• Subtract the \(12x\) term of \(g(x)\) from the zero \(x\) term in \(f(x)\), leaving \(-12x\).
• Subtract the constants, which is \(-5 - (-3)\), simplifying to \(-2\).

Combine these results:\[ (f-g)(x) = x^2 - 12x - 2 \]Function subtraction is useful for understanding how much one function diverges from another over all inputs.

Domain Restrictions

When determining the domain of combined functions like \((f+g)(x)\) or \((f-g)(x)\), understanding domain restrictions is essential. The domain of a function is all the potential input values (x-values) that give a valid output (y-value).
For polynomial functions like \(f(x) = x^2 - 5\) and \(g(x) = 12x - 3\), they are defined for all real numbers because neither function has radicals, denominators that could result in division by zero, nor other restrictions.
Thus, the domain of \((f+g)(x)\) and \((f-g)(x)\) is also all real numbers. In simpler terms, you can plug any real number into these functions without worry of it being undefined or causing breaks in continuity. Recognizing unrestricted domains in polynomials is key when dealing with addition and subtraction of such functions.