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Chapter 2: Problem 63

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x\). $$g(x)=\sqrt{x+3}$$

Short Answer

Expert verified
The inverse of the function is \(g^{-1}(x) = x^2 - 3\).

Step by step solution

01

Understand the Function

The given function is \(g(x) = \sqrt{x + 3}\). This means for each \(x\) input, add 3 and then take the square root to get \(g(x)\). We need to find its inverse function.
02

Swap and Solve for x

To find the inverse, swap the roles of \(x\) and \(y\). Hence, start with \(x = \sqrt{y + 3}\). Square both sides to get \(x^2 = y + 3\) and solve for \(y\) by subtracting 3 from both sides: \(y = x^2 - 3\). Thus, the inverse function is \(g^{-1}(x) = x^2 - 3\).
03

Verify the Inverse

To ensure that \(g(x)\) and its inverse \(g^{-1}(x)\) are correct, check that \(g(g^{-1}(x)) = x\) and \(g^{-1}(g(x)) = x\). Calculate \(g(g^{-1}(x)) = \sqrt{(x^2-3) + 3} = \sqrt{x^2} = |x|\) (consider only positive values for consistency), and \(g^{-1}(g(x)) = (\sqrt{x+3})^2 - 3 = x\).
04

Graph the Function and Inverse

Graph \(g(x) = \sqrt{x+3}\) and \(g^{-1}(x) = x^2 - 3\). Plot the line \(y=x\) on the same graph. Both the function and its inverse should be reflections of each other across this line. Check that every point \((a, b)\) on \(g(x)\) corresponds to \((b, a)\) on \(g^{-1}(x)\).
05

Reflection Verification

Ensure by visual inspection that each graph mirrors the other across the line \(y = x\). This confirms the inverse has been correctly identified.

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Key Concepts

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

One-to-One Functions
Understanding one-to-one functions is key when exploring the concept of inverse functions. A one-to-one function is a type of function where each input maps to a unique output. This means no two different input values will ever produce the same output result. To identify if a function is one-to-one:
  • First, check if it passes the Horizontal Line Test. Draw horizontal lines across the graph of the function. If any line touches the graph at more than one point, it is not a one-to-one function.
  • Additionally, verify uniqueness by checking whether differing domain elements always lead to differing results in the range.
When you have a one-to-one function, you can confidently find its inverse. This is because each unique output from the original function can be clearly reversed back to its specific input, ensuring no ambiguity.
Function Reflections
Function reflections are crucial when comparing a function with its inverse. The graph of a function and its inverse are reflections of each other over the line represented by the equation
\(y = x\). This line acts as a mirror:
  • If you take any point \((a, b)\) on the graph of the original function, its reflection across the line \(y = x\) will be the point \((b, a)\) on the graph of the inverse function.
  • The line \(y = x\) helps us visually verify the correctness of an inverse. On the graph, for every \(x\) in the original function's domain, the corresponding point in the inverse should perfectly mirror across this line.
Checking this reflection is a valuable tool, emphasizing that the operations of converting a function to its inverse - swapping \(x\) and \(y\) and solving - has been done correctly.
Graphing Functions
Graphing functions gives insight into both the behavior and properties of mathematical relations. For the given function \(g(x) = \sqrt{x + 3}\) and its inverse \(g^{-1}(x) = x^2 - 3\), plotting both can visually confirm their inverse relationship.
  • Use graphing tools or software for precise plotting. Start by graphing \(g(x)\) to see the positive square root shape starting at \(x = -3\).
  • Next, graph \(g^{-1}(x)\), noting it should resemble a parabola with its vertex at \((0, -3)\). This shows the parabola's standard form but shifted downward.
  • Finally, draw the line \(y = x\) to check reflection symmetry. The graphs should intersect this line at their corresponding reflections, confirming the inverse relationship.
Graphing functions like these allows you to visually verify important relationships and better understand the dynamics that inverse functions bring to mathematical analysis.

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Most popular questions from this chapter

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