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Chapter 4: Problem 126

Let \(f(x)=\left\\{\begin{array}{ll}x^{2}+2 x-1 & \text { if } x \geq 2 \\ 3 x+1 & \text { if } x<2\end{array}\right.\) Find \(f(5)-f(-5) . \text { (Section } 1.3, \text { Example } 6)\)

Short Answer

Expert verified
The difference \( f(5) - f(-5) = 48 \)

Step by step solution

01

Identify which part of the function to use

Identify which part of the function to use for each of the given \( x \) values. The first \( x \) value is 5. Since 5 is greater than 2, use the first part of the function i.e. \( x^2 + 2x - 1 \). The second \( x \) value is -5. Since -5 is less than 2, use the second part of the function i.e. \( 3x + 1 \).
02

Substitute \( x \) value into appropriate function

Substitute \( x = 5 \) into \( x^2 + 2x - 1 \) to get \( 5^2 + 2*5 - 1 = 34 \). Substitute \( x = -5 \) into \( 3x + 1 \) to get \( 3*(-5) + 1 = -14 \).
03

Subtract the second function value from the first

Subtract the value from the second function from the value from the first function. This gives \( f(5) - f(-5) = 34 - (-14) = 48 \).

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Key Concepts

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

Precalculus
Precalculus is a course that prepares students for calculus, serving as a bridge between algebra, geometry, and calculus itself. It introduces concepts like functions, sequences, series, and the basic understanding of limits. Piecewise functions are a critical topic in precalculus, where a function is defined by different expressions based on the input value.
Function Evaluation
Function evaluation is the process of determining the output of a function for a specific input. In the case of piecewise functions, this involves first determining which part of the function applies to the given input value.

For example, with the function f(x) from the original exercise, you would need to assess the input value against the conditions given for each piece of the function. The evaluation of f(5) and f(-5) requires identifying the correct expression to use for each input.
Substitution Method
The substitution method involves replacing the variable in a function with a given number and simplifying. This is essential for evaluating functions at particular values. Applying the method to our exercise, we input 5 and -5 into their respective expressions of the piecewise function, resulting in two numerical values that can be calculated separately before proceeding to find the difference between them.
Domain of a Function
The domain of a function is the set of all possible input values (typically represented as 'x') that the function can accept without leading to any undefined or non-real numbers. In piecewise functions, the domain is segmented based on the conditions that define each piece.

In the exercise provided, there are two separate domains: one where x is greater than or equal to 2, and the other where x is less than 2. Identifying the domain is crucial for accurately evaluating the function.

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