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Chapter 10: Problem 38

For each function, find the specified function value, if it exists. If it does not exist, state this. $$ g(x)=-\sqrt{(x+1)^{2}} ; g(-3), g(4), g(-5) $$

Short Answer

Expert verified
g(-3) = -2, g(4) = -5, g(-5) = -4

Step by step solution

01

Understand the function

The given function is \[ g(x) = - \sqrt{(x+1)^2} \]. We need to find the values of this function at specific points: \( g(-3) \), \( g(4) \), and \( g(-5) \).
02

Simplify the function

Let's simplify the function first. Notice that \( (x+1)^2 \) is always non-negative and its square root is \( |x+1| \). Therefore, the function becomes:\[ g(x) = -|x+1| \]
03

Evaluate \( g(-3) \)

To find \( g(-3) \), substitute \( x = -3 \) into the simplified function:\[ g(-3) = -|-3+1| = -|-2| = -2 \]
04

Evaluate \( g(4) \)

To find \( g(4) \), substitute \( x = 4 \) into the simplified function:\[ g(4) = -|4+1| = -|5| = -5 \]
05

Evaluate \( g(-5) \)

To find \( g(-5) \), substitute \( x = -5 \) into the simplified function:\[ g(-5) = -|-5+1| = -|-4| = -4 \]

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Key Concepts

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

Understanding Absolute Value
Absolute value is a fundamental concept in mathematics that measures how far a number is from zero on a number line. Whether the number is positive or negative, its absolute value is always non-negative. For instance, \(|-3| = 3\) because \-3\ is 3 units away from zero. Similarly, \(|3| = 3\). This property is very useful when dealing with functions that involve absolute values, as it guarantees a non-negative result regardless of input.
Function Simplification
Simplifying functions is a powerful technique that can make them easier to work with. It involves rewriting the function in a simpler, more convenient form while retaining its original properties. In our exercise, the function given is \(- \sqrt{(x+1)^2}\). We can recognize that the term \((x+1)^2\) under the square root is always non-negative. Applying the square root results in \(|x+1|\), the absolute value of \(x+1\). Hence, the function simplifies to \(-|x+1|\). Simplifying functions makes it easier to evaluate them at specific points and can reveal insights about their behavior.
Substitution in Functions
Substitution is a straightforward yet crucial concept in evaluating functions. It involves replacing the variable in the function with a specific value. This lets you determine the output for given inputs. For example, to evaluate \(g(x) = -|x+1|\) at \(-3\), simply replace \(x\) with \(-3\): \(|-3+1| = |-2| = 2\). Thus, \(g(-3) = -2\). Repeat the process for other values like \(4\) and \(-5\) to find \(g(4) = -5\) and \(g(-5) = -4\). Substitution is essential for determining specific function values and is widely used in various mathematical applications.

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