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

Ch. 2 - Graphs and Functions

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Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 58

Chapter 3, Problem 58

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

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Identify the function g(x) given as \(g(x) = -x^{2} + 4x + 1\).

Substitute the value \(x = -\frac{1}{4}\) into the function \(g(x)\), so write \(g\left(-\frac{1}{4}\right) = -\left(-\frac{1}{4}\right)^{2} + 4\left(-\frac{1}{4}\right) + 1\).

Calculate the square of \(-\frac{1}{4}\), which is \(\left(-\frac{1}{4}\right)^{2} = \frac{1}{16}\), and substitute it back into the expression.

Multiply each term: \(-\left(\frac{1}{16}\right)\), \(4 \times \left(-\frac{1}{4}\right)\), and keep the constant \(+1\) as is.

Combine all the terms by adding and subtracting to simplify the expression and find the value of \(g\left(-\frac{1}{4}\right)\).

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Key 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 g(-1/4), replace every x in g(x) with -1/4 and simplify the expression.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Understanding how to handle terms like -x² and 4x is essential for correctly evaluating g(x) at a specific value.

Simplification of Algebraic Expressions

Simplification involves performing arithmetic operations and combining like terms to write expressions in their simplest form. After substituting values into functions, simplifying ensures the final answer is clear and concise.

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