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Magazine Chapter 5: Problem 33
For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=(x+3)^{2}, \quad x \geq-3 $$
Short Answer
Expert verified
The inverse function is \( f^{-1}(x) = \sqrt{x} - 3 \).
Step by step solution
01
Understand the Inverse
To find the inverse of a function, you need to interchange the roles of "x" and "y" in the function's equation. If the function is given as \( y = f(x) \), then to find its inverse, solve for \( x \) in terms of \( y \).
02
Replace f(x) with y
Replace \( f(x) \) with \( y \) to make it easier to find the inverse. The given function is \( y = (x+3)^2 \).
03
Swap x and y.
Interchange \( x \) and \( y \) to set up the equation for the inverse function: \( x = (y + 3)^2 \).
04
Solve for y
To isolate \( y \), take the square root of both sides of the equation, considering the given domain \( x \geq -3 \): \( y + 3 = \sqrt{x} \). Because of the domain limitation in the original function, we use only the positive root for \( x \), resulting in \( y = \sqrt{x} - 3 \).
05
Express the Inverse Function
After solving for \( y \), you can now express the inverse function as \( f^{-1}(x) = \sqrt{x} - 3 \).
06
Graphing the Function and its Inverse
Graph the original function, \( f(x) = (x+3)^2 \), which is a parabola with vertex at \((-3, 0)\) opening upwards. The inverse function \( f^{-1}(x) = \sqrt{x} - 3 \) is a half-parabola with the domain \( x \geq 0 \), shifted 3 units down. The line of symmetry (\( y = x \)) should also be drawn as both graphs should be mirror images about this line.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Graphing
Graphing functions enables us to visualize their behavior. When a function and its inverse are plotted, the inverse is a reflection of the original function over the line \(y = x\). For all functions:
- The x-coordinates become the y-coordinates in their inverse.
- Graph both the function and its inverse on the same set of axes for easy comparison.
- The line \( y = x \) is a significant line of symmetry, as it visually represents this relationship.
Parabola
A parabola is a U-shaped graph that is characteristic of quadratic functions, like \( f(x) = (x+3)^2 \). It is a symmetrical graph, typically having a vertex, which is the highest or lowest point depending on its orientation.
The general formula for a quadratic function is \( ax^2 + bx + c \). In our case, the graph is shifted:
To graph, plot points considering this restriction. Its symmetry about the vertical line through the vertex is important for sketching accurately.
The general formula for a quadratic function is \( ax^2 + bx + c \). In our case, the graph is shifted:
- To the left, by 3 units, due to \((x+3)^2\).
- Upwards or downwards, which can be identified by the constant term, but here the vertex \((-3,0)\) lies on the x-axis.
To graph, plot points considering this restriction. Its symmetry about the vertical line through the vertex is important for sketching accurately.
Domain and Range
The domain of a function is all the possible input values (x-values). The range is all the possible output values (y-values). For example, the function \( f(x) = (x+3)^2 \) has a domain of \( x \geq -3 \) and a range of \( y \geq 0 \), since squaring any real number results in a non-negative value.
For inverse functions, these roles switch. So:
- Restrictions on the domain affect the range of the inverse function.
- Inverse relations are mirrored in domain and range but may only cover parts of the original set.
For inverse functions, these roles switch. So:
- The domain of the inverse function \( f^{-1}(x) = \sqrt{x} - 3 \) is \( x \geq 0 \).
- The range is \( y \geq -3 \), reflecting the y-values of the original function's x-values.
- Restrictions on the domain affect the range of the inverse function.
- Inverse relations are mirrored in domain and range but may only cover parts of the original set.
Square Root Function
Square root functions, such as \( f^{-1}(x) = \sqrt{x} - 3 \), are derived from taking the square root of the input value. They have characteristics worth noting:
When examining the square root function:
- They only include non-negative values in their domain because the square root of a negative number isn't real.
- The graph starts at a specific point, which here is \((0, -3)\), then increases as x increases.
When examining the square root function:
- Consider both the graph and its transformations.
- Graph points starting from the adjusted origin to sketch accurately.
