Textbook Question
Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y = √(4x + 1)
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0. Review of College Algebra
Functions
1:24 minutesProblem 49
Textbook Question
Let f(x) = -3x + 4 and g(x) = -x² + 4x + 1. Find each of the following. Simplify if necessary. See Example 6. ƒ(⅓)
Verified step by step guidance1
Identify the function f(x) given as \(f(x) = -3x + 4\).
Substitute the value \(x = \frac{1}{3}\) into the function \(f(x)\) to find \(f\left(\frac{1}{3}\right)\).
Write the substitution explicitly: \(f\left(\frac{1}{3}\right) = -3 \times \frac{1}{3} + 4\).
Simplify the multiplication part: \(-3 \times \frac{1}{3} = -1\).
Add the results to get the simplified expression: \(-1 + 4\).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Evaluation
Function evaluation involves substituting a given input value into the function's formula and simplifying to find the output. For example, to find ƒ(⅓), replace x with ⅓ in the expression for ƒ(x) and simplify the result.
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Polynomial Functions
Polynomial functions are expressions involving variables raised to whole number powers with coefficients. Understanding how to manipulate and simplify polynomials, such as -x² + 4x + 1, is essential for evaluating functions like g(x).
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Simplification of Algebraic Expressions
Simplification involves combining like terms and performing arithmetic operations to write expressions in their simplest form. This skill is crucial when evaluating functions to ensure the final answer is clear and concise.
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