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Chapter 1: Problem 47

Evaluate the function for the given value of \(x\). \(f(x)=4 x-1 ; x=-1\)

Short Answer

Expert verified
\(f(-1) = -5\)

Step by step solution

01

Identify Function and Value of x

The function given in the problem is \(f(x)=4x-1\), and the given value for \(x\) is -1.
02

Substitute Value of x

Substitute \(x=-1\) into the function, so it will become \(f(-1)=4(-1)-1\).
03

Calculate Result

Now simplify the expression: \(f(-1)= -4 - 1 = -5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substituting Values in Functions
Understanding how to substitute values into functions is a fundamental skill in algebra. When you are given a function, like the one in the exercise, which is written as f(x) = 4x - 1, you're dealing with a rule that tells you how to calculate the output ('f(x)') for a given input ('x').

When you substitute a specific value for x, in this case, -1, you replace every occurrence of x in the function with -1. This leads to a new equation: f(-1) = 4(-1) - 1, which is an application of this fundamental concept. Through substitution, you convert the abstract formula into a concrete numerical expression that can be evaluated step by step to find the result.
Simplifying Expressions
Simplifying expressions is a process of rewriting them in a more understandable or 'simpler' form while keeping the original value unchanged. This usually involves combining like terms, using arithmetic operations, and eliminating unnecessary parts of the expression to make it clearer or easier to work with.

In our exercise, after substituting -1 for x, we end up with f(-1) = 4(-1) - 1. To simplify, we multiply 4 by -1 to get -4 and then subtract 1 to arrive at the simplified result of -5. Simplifying expressions is an essential step in function evaluation as it helps in accurately finding the output value for any given input value.
Function Operations
Function operations refer to the manipulation and evaluation of functions using arithmetic operations such as addition, subtraction, multiplication, and division. The execution of these operations follows the order of operations (PEMDAS/BODMAS) to ensure the correct result.

In our example, the operation takes the form of f(-1) = 4(-1) - 1, where multiplication comes first, followed by subtraction. Completing these operations yields the final value of the function for x = -1, which is -5. Mastering function operations allows students to correctly evaluate expressions and solve more complex equations involving functions.

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Most popular questions from this chapter

\(\frac{1}{2} x+4 y=4\) \(2 x-y=1\)

OPEN-ENDED Give two real-life quantities that have (a) a positive correlation, (b) a negative correlation, and (c) approximately no correlation. Explain.

Find the \(x\)-intercept and the \(y\)-intercept of the graph of the equation. \(y=x+2\)

Use a graphing calculator to graph the function and its parent function. Then describe the transformations. \(f(x)=\frac{3}{4}|x|+1\)

A florist must make 5 identical bridesmaid bouquets for a wedding. The budget is \(\$ 160\), and each bouquet must have 12 flowers. Roses cost \(\$ 2.50\) each, lilies cost \(\$ 4\) each, and irises cost \(\$ 2\) each. The florist wants twice as many roses as the other two types of flowers combined. a. Write a system of equations to represent this situation, assuming the florist plans to use the maximum budget. b. Solve the system to find how many of each type of flower should be in each bouquet. c. Suppose there is no limitation on the total cost of the bouquets. Does the problem still have exactly one solution? If so, find the solution. If not, give three possible solutions.
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