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Chapter 9: Problem 52

Find the inverse of each function. $$g(x)=(x+5)^{2} \text { for } x \geq-5$$

Short Answer

Expert verified
The inverse function is \( g^{-1}(x) = \sqrt{x} - 5 \).

Step by step solution

01

Understanding the Function

Analyze the given function: \[ g(x) = (x+5)^{2} \text{ for } x \text{ greater than or equal to } -5 \] Note that the function is defined for \( x \geq -5 \). This means the function has a domain restriction.
02

Replace \( g(x) \) with \( y \)

Write the function as: \[ y = (x+5)^{2} \]
03

Solve for \( x \)

To find the inverse, solve the equation for \( x \): \[ y = (x+5)^{2} \] Take the square root of both sides: \[ \sqrt{y} = x+5 \] Since \( x \geq -5 \), we consider the positive root: \[ x + 5 = \sqrt{y} \] Subtract 5 from both sides: \[ x = \sqrt{y} - 5 \]
04

Replace \( y \) with \( x \)

Replace \( y \) with \( x \) to write the inverse function: \[ g^{-1}(x) = \sqrt{x} - 5 \]

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Key Concepts

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

domain restrictions
A function's domain is the set of all possible input values (usually denoted as \( x \)) that the function can accept. In the given problem, the function \( g(x) = (x+5)^2 \) has a specific domain restriction: \( x \geq -5 \). This restriction is crucial because it impacts the inverse function as well. Domain restrictions often arise due to the nature of the function. For example, polynomials involving square roots naturally limit the domain to non-negative numbers, since the square root of a negative number is not a real number. Understanding the domain helps ensure that the function behaves as expected and avoids undefined values.
square roots
Square roots are the opposite operation of squaring a number. In other words, if \( y = a^2 \), then \( a = \sqrt{y} \). When working with square roots in equations, remember two key points:
  • There are always two roots for any positive number, a positive root and a negative root, because both \((a)\) and \((-a)\) when squared, give \(a^2\).
  • However, due to the domain restriction \(x \geq -5\), we only consider the principal (positive) root in this problem.
The concept of square roots is utilized in step 3 of the solution, when both sides of the equation \( y = (x+5)^2 \) are square-rooted to isolate \( x \) on one side.
solving equations
Solving equations involves finding the value of the variable that makes the equation true. Here, we start from the function \(y = (x+5)^2\). To solve for \(x\), follow these steps:
  • Take the square root of both sides to get \( \sqrt{y} = x+5 \).
  • Due to the domain restriction, we consider only the positive root. This simplifies to \( x = \sqrt{y} - 5 \).
Rewriting this equation after isolating \( x \) gives us the inverse function. These steps are common in solving for variables within algebraic equations. Make sure to systematically rearrange terms and apply the appropriate operations to both sides of the equation to keep it balanced.

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Most popular questions from this chapter

Classify each function as either a linear, constant, quadratic, square-root, or absolute value function. $$f(x)=-3$$

Graph \(y=x^{2}, y=(x+5)^{2},\) and \(y=(x-2)^{2}\) on the same screen. What can you say about the position of \(y=(x-h)^{2}\) relative to \(y=x^{2} ?\)

Solve each problem. Sail area-displacement ratio. To find the sail areadisplacement ratio \(S,\) first find \(y,\) where \(y=(d / 64)^{2 / 3}\) and \(d\) is the displacement in pounds. Next find \(S,\) where \(S=A / y\) and \(A\) is the sail area in square feet. a) For the Pacific Seacraft \(40, A=846\) square feet \(\left(\mathrm{ft}^{2}\right)\) and \(d=24,665\) pounds. Find \(S\) b) For a boat with a sail area of \(900 \mathrm{ft}^{2},\) write \(S\) as a function of \(d\) c) For a fixed sail area, does \(S\) increase or decrease as the displacement increases?

If the graph of \(y=x^{2}\) is translated eight units upward, then what is the equation of the curve at that location?

Graph each function and state the domain and range. $$y=-x^{2}+4 x-4$$
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