Let ƒ(x)=x²+3 and g(x)=-2x+6. Find each of the following. (ƒ-g)(-1)

Verified step by step guidance

1. Understand that the expression (ƒ - g)(x) means you subtract the function g(x) from the function ƒ(x). So, (ƒ - g)(x) = ƒ(x) - g(x).
2. Write down the given functions: ƒ(x) = x^2 + 3 and g(x) = -2x + 6.
3. Substitute the expressions for ƒ(x) and g(x) into the difference: (ƒ - g)(x) = (x^2 + 3) - (-2x + 6).
4. Simplify the expression by distributing the negative sign across g(x): (ƒ - g)(x) = x^2 + 3 + 2x - 6.
5. Now, evaluate (ƒ - g)(-1) by substituting x = -1 into the simplified expression: (ƒ - g)(-1) = (-1)^2 + 3 + 2(-1) - 6.

Key Concepts

Function Notation and Evaluation

Function notation, such as ƒ(x), represents a rule that assigns each input x to an output value. Evaluating a function at a specific value means substituting that value into the function's expression and simplifying to find the output.

Function Operations (Addition and Subtraction)

Function operations involve combining two functions by adding or subtracting their outputs for the same input. For example, (ƒ - g)(x) means subtracting g(x) from ƒ(x) and simplifying the resulting expression before evaluating.

Substitution and Simplification

After forming the combined function expression, substitution involves replacing the variable with a given number, such as -1. Simplification then reduces the expression to a single numerical value, which is the final answer.