Let ƒ(x)=-3x+4 and g(x)=-x²+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(x+2)

Verified step by step guidance

1. Identify the given functions: \( f(x) = -3x + 4 \) and \( g(x) = -x^2 + 4x + 1 \). We are asked to find \( f(x+2) \), which means we need to evaluate the function \( f \) at \( x+2 \) instead of \( x \).
2. To find \( f(x+2) \), substitute \( x+2 \) into the function \( f(x) \) wherever you see \( x \). So, replace \( x \) with \( x+2 \) in the expression \( -3x + 4 \).
3. Write the substitution explicitly: \( f(x+2) = -3(x+2) + 4 \). This means multiply \( -3 \) by the entire quantity \( (x+2) \) and then add 4.
4. Distribute the \( -3 \) across the terms inside the parentheses: \( -3 \times x = -3x \) and \( -3 \times 2 = -6 \). So, \( f(x+2) = -3x - 6 + 4 \).
5. Combine like terms: \( -6 + 4 = -2 \). Therefore, the simplified expression for \( f(x+2) \) is \( -3x - 2 \).

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 ƒ(x+2) means substituting x+2 into the function in place of x, then simplifying the expression to find the output.

Polynomial Expressions and Simplification

Polynomials are algebraic expressions involving variables raised to whole-number exponents. Simplifying polynomial expressions involves combining like terms and performing arithmetic operations to write the expression in its simplest form.

Composition of Functions

Composition involves applying one function to the result of another, often written as ƒ(g(x)) or ƒ(x+2). Understanding how to substitute expressions correctly is essential for evaluating composite functions or function values at shifted inputs.