For the pair of functions defined, find (ƒ+g)(x).Give the domain of each. See Example 2.
ƒ(x)=3x+4, g(x)=2x-5

Verified step by step guidance

1. To find (ƒ+g)(x), we need to add the two functions ƒ(x) and g(x) together. This means we write (ƒ+g)(x) = ƒ(x) + g(x).
2. Substitute the given functions into the expression: (ƒ+g)(x) = (3x + 4) + (2x - 5).
3. Combine like terms by adding the coefficients of x and the constant terms separately: (3x + 2x) + (4 - 5).
4. Simplify the expression to get the combined function: (ƒ+g)(x) = 5x - 1.
5. Determine the domain of each function. Since both ƒ(x) = 3x + 4 and g(x) = 2x - 5 are linear functions, their domains are all real numbers, so the domain of (ƒ+g)(x) is also all real numbers.

Key Concepts

Function Addition

Function addition involves creating a new function by adding the outputs of two given functions for the same input value. For functions ƒ(x) and g(x), (ƒ+g)(x) = ƒ(x) + g(x). This operation combines the expressions algebraically to form a single function.

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. When combining functions, the domain of the resulting function is the intersection of the domains of the original functions, ensuring all inputs are valid for both.

Linear Functions

Linear functions have the form f(x) = mx + b, where m and b are constants. They are defined for all real numbers, so their domain is typically all real numbers unless otherwise restricted. Understanding this helps in determining the domain of combined linear functions.