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

Ch. 2 - Graphs and Functions

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Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 66

Chapter 3, Problem 66

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

Verified step by step guidance

Identify the given functions: $f(x) = -3x + 4$ and $g(x) = -x^{2} + 4x + 1$.

Focus on the function $f(x)$ since the problem asks for $f(3t - 2)$, which means we will substitute $x$ in $f(x)$ with the expression $3t - 2$.

Write the substitution explicitly: $f(3t - 2) = -3(3t - 2) + 4$.

Apply the distributive property to multiply $-3$ by each term inside the parentheses: $-3 \times 3t$ and $-3 \times (-2)$.

Simplify the expression by combining like terms after distribution, but do not calculate the final numeric value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Notation and Evaluation

Function notation, such as f(x), represents a rule that assigns each input x to an output f(x). Evaluating a function means substituting the given expression or value into the function's formula and simplifying the result.

Substitution of Algebraic Expressions

Substitution involves replacing the variable in a function with an algebraic expression instead of a single number. This requires careful distribution and simplification to correctly evaluate the function for the given input.

Simplifying Algebraic Expressions

Simplifying expressions means combining like terms and performing arithmetic operations to write the expression in its simplest form. This step ensures the final answer is clear and concise.

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