Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ+g)(-5)

Verified step by step guidance

1. Identify the functions: $f(x)=x^2+3$ and $g(x)=-2x+6$.
2. Understand that $(f+g)(x)$ means adding the two functions: $(f+g)(x)=f(x)+g(x)$.
3. Substitute the expressions for $f(x)$ and $g(x)$ into the equation: $(f+g)(x)=(x^2+3)+(-2x+6)$.
4. Simplify the expression by combining like terms: $(f+g)(x)=x^2-2x+9$.
5. Evaluate $(f+g)(-5)$ by substituting $-5$ into the simplified expression: $(-5{)}^2-2(-5)+9$.

Key Concepts

Function Addition

Function addition involves combining two functions by adding their outputs for the same input. If ƒ(x) and g(x) are two functions, then (ƒ+g)(x) is defined as ƒ(x) + g(x). This operation allows us to create a new function that represents the sum of the original functions.

Evaluating Functions

Evaluating a function means substituting a specific value into the function's equation to find the output. For example, to evaluate (ƒ+g)(-5), we first calculate ƒ(-5) and g(-5) separately, then add the results together. This process is essential for finding the value of the combined function at a given point.

Quadratic and Linear Functions

Quadratic functions, like ƒ(x) = x^2 + 3, have a parabolic shape and are defined by the highest exponent of 2, while linear functions, like g(x) = -2x + 6, represent straight lines with a constant rate of change. Understanding the characteristics of these functions is crucial for performing operations like addition and for visualizing their graphs.