Function inverses example 2 | Functions and their graphs | Algebra II | Khan Academy

Khan Academy
16 Jul 201007:12
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TLDRThe video script explains the process of finding the inverse of a given function, f(x) = (x + 2)^2 + 1, with a domain constraint of x ≥ -2. It emphasizes the importance of domain constraints in determining the correct inverse function. The script guides viewers through algebraic steps, including subtracting 1, taking the square root, and considering the correct sign based on the domain. It also introduces the concept of the range of y-values and how it affects the inverse function's domain. The final inverse function is presented as f^(-1)(x) = √(x - 1) - 2 for x ≥ 1, with a graphical representation showing the relationship between the original function and its inverse as mirror images across the line y = x.

Takeaways
  • 🔍 The function \( f(x) = (x + 2)^2 + 1 \) is being discussed, with a domain constraint of \( x \geq -2 \).
  • 📚 The domain constraint is necessary because the function's inverse requires a one-to-one mapping, which is not possible with the full parabola.
  • 🤔 The instructor prompts viewers to consider why the domain was constrained and hints at a future video addressing this.
  • 🔄 The process of finding the inverse function involves swapping \( x \) and \( y \) and solving for \( x \) in terms of \( y \).
  • ➖ The first step in finding the inverse is to subtract 1 from both sides, resulting in \( y - 1 = (x + 2)^2 \).
  • 📏 The square root is taken next, with an emphasis on considering the correct sign (positive in this case) due to the domain constraint.
  • 🌐 The domain of the original function is \( x \geq -2 \), which implies that \( x + 2 \geq 0 \), thus the positive square root is taken.
  • 📉 The range of the original function is determined to be \( y \geq 1 \), which becomes the domain for the inverse function.
  • 🔄 After solving for \( x \), the inverse function is expressed as \( f^{-1}(y) = \sqrt{y - 1} - 2 \) for \( y \geq 1 \).
  • 🔠 The inverse function is then rewritten with \( x \) as the input, resulting in \( f^{-1}(x) = \sqrt{x - 1} - 2 \) for \( x \geq 1 \).
  • 📈 The graph of the inverse function is a reflection of the original function's graph across the line \( y = x \), illustrating the concept of inverse functions.
Q & A

  • What is the function f(x) described in the script?

    -The function f(x) is defined as f(x) = (x + 2)^2 + 1, with the domain constraint that x must be greater than or equal to -2.

  • Why is the domain of x constrained to be greater than or equal to -2?

    -The domain is constrained to ensure that the expression (x + 2) is always positive, which is necessary for taking the square root in the correct domain when finding the inverse function.

  • What is the purpose of finding the inverse of a function?

    -The purpose of finding the inverse of a function is to determine a new function that reverses the input-output relationship of the original function, effectively mapping y-values back to x-values.

  • How does the script suggest to start finding the inverse of the given function?

    -The script suggests starting by setting y equal to the function expression and then solving for x in terms of y.

  • What is the first step in solving for x in terms of y?

    -The first step is to subtract 1 from both sides of the equation, resulting in y - 1 = (x + 2)^2.

  • Why is it important to consider the positive or negative square root when solving for x?

    -It is important because the domain constraint on x dictates that (x + 2) must be non-negative, so the positive square root should be taken to ensure the correct domain for the inverse function.

  • What is the domain constraint for the inverse function based on the original function's range?

    -The domain constraint for the inverse function is that y must be greater than or equal to 1, as the original function's range starts from this value.

  • How does the script guide us to express the inverse function in terms of x?

    -The script guides us to express the inverse function by renaming y with x, resulting in the inverse function f^(-1)(x) = sqrt(x - 1) - 2 for x ≥ 1.

  • What is the significance of the points (1, -2), (2, -1), and (5, 0) in the context of the inverse function?

    -These points are examples of the inverse function's graph, showing the mapping of y-values to x-values based on the inverse function formula.

  • How does the script describe the relationship between the original function and its inverse graphically?

    -The script describes the relationship as mirror images around the line y = x, indicating that the input and output have been swapped in the inverse function.

Outlines
00:00
📚 Finding the Inverse of a Constrained Function

This paragraph discusses the process of finding the inverse of the function f(x) = (x + 2)^2 + 1, which is defined for x ≥ -2. The video script explains the importance of constraining the domain to ensure the function is one-to-one, a prerequisite for finding an inverse. It proceeds by illustrating the step-by-step algebraic manipulation to solve for x in terms of y, which involves subtracting 1 from both sides and taking the square root, considering the positive root due to the domain constraint. The script also introduces the concept of the range of y-values and establishes that y ≥ 1 for the given domain of x. The process concludes with rewriting the equation to express x in terms of y, effectively finding the inverse function.

05:03
📈 Graphing the Inverse Function and Understanding Its Domain

The second paragraph delves into graphing the inverse function of the previously discussed function. It begins by renaming y with x to represent the inverse function as f^(-1)(x) = √(x - 1) - 2 for x ≥ 1. The script then attempts to graph the inverse function by plotting points that correspond to the minimum value x can take and other selected x-values. The points (1, -2), (2, -1), and (5, 0) are calculated and plotted to give a visual representation of the inverse function's graph. The paragraph concludes by noting that the inverse graph is a mirror image of the original function's graph across the line y = x, highlighting the symmetry that occurs when x and y are swapped in the context of function inverses.

Mindmap
Keywords
💡Function
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In the context of the video, the function f(x) = (x + 2)^2 + 1 is the primary focus, where x is the input and f(x) is the output. The video discusses finding the inverse of this function, which is a process of reversing the roles of inputs and outputs.
💡Domain
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In the video, the domain is specifically mentioned as x being greater than or equal to negative 2, which is a constraint placed on the original function to ensure it has an inverse.
💡Inverse Function
An inverse function reverses the effect of the original function. It maps each output of the original function to its corresponding input. The video's main theme revolves around finding the inverse of the given function f(x), which is denoted as f^(-1)(x) or f^(-1)(y) in different contexts.
💡Square
The term 'squared' in the video script refers to the mathematical operation of squaring a number, which is multiplying the number by itself. In the function f(x) = (x + 2)^2 + 1, '(x + 2) squared' means (x + 2) * (x + 2), and this operation is a key part of the function that needs to be addressed when finding its inverse.
💡Square Root
The square root operation is the inverse of squaring a number. It finds a value that, when multiplied by itself, gives the original number. In the video, the square root is used to undo the squaring in the function when finding the inverse, as seen in the step where '√(y - 1)' is derived from 'y - 1 = (x + 2)^2'.
💡Positive Square Root
When taking the square root of a squared term, there are two possible results: the positive and the negative root. The video emphasizes choosing the positive square root due to the domain constraint of the original function, which ensures that the expression (x + 2) is always non-negative.
💡Range
The range of a function is the set of all possible output values (y-values). In the video, it is determined that the range of the original function is y ≥ 1, which becomes the domain of the inverse function because the roles of x and y are swapped.
💡Mapping
In the context of functions, mapping refers to the process of assigning each input value to exactly one output value. The video discusses the concept of mapping in the context of finding the inverse function, where the direction of the mapping is reversed.
💡Graph
A graph in mathematics is a visual representation of the relationship between inputs and outputs of a function. The video discusses the graphical representation of both the original function and its inverse, highlighting how they are mirror images across the line y = x.
💡Mirror Images
In the context of the video, 'mirror images' refers to the symmetrical relationship between the graph of a function and its inverse when reflected across the line y = x. This concept is used to illustrate the visual outcome of finding an inverse function.
Highlights

Introduction of the function f(x) = (x + 2)^2 + 1 with a constrained domain of x ≥ -2.

The necessity of constraining the domain to find the inverse function is discussed.

Exploration of why the full parabola might not allow for an inverse function.

The process of finding the inverse function by solving for x in terms of y.

Subtracting 1 from both sides to isolate the squared term.

The importance of choosing the correct square root (positive) based on the domain constraints.

Taking the positive square root to maintain the domain of x ≥ 0.

Determining the range of y-values, which is y ≥ 1, from the function's graph.

The significance of the y constraint for the inverse function's domain.

Solving for x in terms of y by subtracting 2 from both sides.

Expressing the inverse function in terms of y with the domain y ≥ 1.

Renaming y with x to express the inverse function as f^(-1)(x).

The inverse function f^(-1)(x) = √(x - 1) - 2 for x ≥ 1.

Graphing the inverse function and identifying points on the graph.

The inverse graph is only defined for x ≥ 1, unlike the original function.

The relationship between the original function and its inverse as mirror images around the line y = x.

Visual demonstration of how the points are mirrored around the line y = x.

Transcripts

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Inverse FunctionsDomain ConstraintsMathematicsSquare RootFunction MappingGraph AnalysisEducational ContentMath TutorialAlgebra ConceptsInverse Graphing