Let $\text{ƒ}\left(x\right)=x^2+3$ and $g\left(x\right)=-2x+6$. Find each of the following. See Example 1.
$(\text{ƒ}+g)(-5)$

Verified step by step guidance

1. Understand that (ƒ+g)(x) means you add the functions ƒ(x) and g(x) together, so (ƒ+g)(x) = ƒ(x) + g(x).
2. Write the expressions for ƒ(x) and g(x): ƒ(x) = x^2 + 3 and g(x) = -2x + 6.
3. Add the two functions together: (ƒ+g)(x) = (x^2 + 3) + (-2x + 6).
4. Simplify the expression by combining like terms: (ƒ+g)(x) = x^2 - 2x + (3 + 6).
5. Evaluate the simplified expression at x = -5 by substituting -5 for x: (ƒ+g)(-5) = (-5)^2 - 2(-5) + 9.

Key Concepts

Function Notation and Evaluation

Function notation, such as ƒ(x), represents a rule that assigns each input x to an output. Evaluating a function means substituting a specific value for x and calculating the result. For example, ƒ(-5) means replacing x with -5 in the function ƒ(x).

Function Addition

The sum of two functions (ƒ + g)(x) is defined as ƒ(x) + g(x). To find (ƒ + g)(-5), evaluate each function at -5 separately, then add the results. This operation combines the outputs of both functions for the same input.

Polynomial and Linear Functions

ƒ(x) = x² + 3 is a polynomial function of degree 2, and g(x) = -2x + 6 is a linear function. Understanding their forms helps in correctly substituting values and performing arithmetic operations on their outputs.