6D - Inverse functions
Defining the inverse of a function:
In mathematics you have already encountered inverse operations such as addition and subtraction, or multiplication and division. In both of these instances, one operation 'undoes' the other. We can define functions and their inverse functions in a similar way.
In mathematics you have already encountered inverse operations such as addition and subtraction, or multiplication and division. In both of these instances, one operation 'undoes' the other. We can define functions and their inverse functions in a similar way.
An algebraic definition of an inverse function:
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A geometric definition of an inverse function:
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Activity: Drawing the inverse of a function
Insert instructions and sample work here.
Determining the inverse of a function:
All functions have an inverse relation; however, on a specific type of function has an inverse function. For an inverse function, \(f^{-1}\), to exist the original function, \(f\), must be a one-to-one function.
All functions have an inverse relation; however, on a specific type of function has an inverse function. For an inverse function, \(f^{-1}\), to exist the original function, \(f\), must be a one-to-one function.
To determine the rule for an inverse function:
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■ Important: To fully define an inverse function you must specify the rule and the domain. ■ Remember: The domain of \(f^{-1}\) is equal to the range of \(f\). |
6D - Example 1: Finding the inverse of a function
Fully define the inverse of the function \(f:R→R, f(x)=2x+3\).
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6D - Example 1: Video solution
6D - Example 1: Practice
Question 1: If \(f(x)=3x\), find the rule for \(f^{-1}(x)\). Question 2: If \(g(x)=2-5x\), find the inverse function, \(g^{-1}\). 6D - Example 1: Solutions
Question 1: \[f^{-1}(x)=\frac{x}{3}\] Question 2: \[g^{-1}:R→R, g^{-1}(x)=\frac{-x+2}{5}\]
■ The instructions "find \(f^{-1}(x)\)" generally implies find the rule of the inverse function. However, "find \(f^{-1}\)" generally requires you to state the rule and the domain of the inverse function. |
6D - Example 2: Finding the inverse of a function
Fully define the inverse of the function \(f(x)=x^2-6\), where \(x\in (-∞, 0]\).
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6D - Example 2: Video solution
6D - Example 2: Practice
Question 1: If \(f(x)=(x+3)^2\), where \(x \geq -3\), find \(f^{-1}\). Question 2: If \(g:[-4, \infty)→R, g(x)=- \sqrt{x+4}-1\), find \(g^{-1}\). 6D - Example 2: Solutions
Question 1: \(f^{-1}(x)=\sqrt{x}-3\), where \(x \geq 0\). Question 2: \[g^{-1}:(-\infty,-1]→R, g^{-1}(x)=(x+1)^2-4\] |
Determining the points of intersection between a function and its inverse:
Previously, we noted that a function, \(f\), and its inverse, \(f^{-1}\), intersect along the line \(y=x\). Therefore, to find the points of intersection between \(f(x)\) and \(f^{-1}(x)\) you can solve any of the following three equations:
Previously, we noted that a function, \(f\), and its inverse, \(f^{-1}\), intersect along the line \(y=x\). Therefore, to find the points of intersection between \(f(x)\) and \(f^{-1}(x)\) you can solve any of the following three equations:
- \(f(x)=f^{-1}(x)\)
- \(f(x)=x\)
- \(f^{-1}(x)=x\)
6D - Example 3: Finding the points of intersection between a function and its inverse
Consider the function, graphed below, that is defined by \[f(x)=\frac{2}{x+2}\]
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6D - Example 2: Video solution
6D - Example 2: Practice
Question 1:
Consider the function \(f(x)=(x-2)^2\), where \(x \geq 2\).
Question 2: Show that the function with rule \(g(x)=\frac{6}{x+2}-1\) intersects its inverse function, \(g^{-1}\) at the coordinates \((-4, -4)\) and \((1, 1)\). 6D - Example 2: Solutions
Question 1: (a) \(f^{-1}(x)=\sqrt{x}+2\) (b) \(f(x)=x \therefore (4,4)\) Question 2: \[x=\frac{6}{x+2}-1\] \[\therefore (x+1)(x+2)=6\] \[\therefore x^2+3x+2=6\] \[\therefore x^2+3x-4=0\] \[\therefore (x+4)(x-1)=0\] \[\therefore x=-4, x=1\] As the points of intersection occur along the line \(y=x\) the coordinates are \((-4, -4)\) and \((1, 1)\). |
Thinking with parameters: How many points of intersection?
Consider the function \(f(x)=x^2+c\), where \(c \in R\). Find the value(s) of \(c\) such that the function, \(f\), and its inverse relation, \(f^{-1}\), intersect at least once. You may wish to use the interactive graph below to explore the problem by changing the value of \(c\) using the slider.
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Video solution:
Coming soon... Solutions:
To find the points of intersection between a function and its inverse, we can solve the equation \(f(x)=x\). \[\therefore x^2+c=x\] \[\therefore x^2-x+c=0\] The discriminant , \(\Delta = b^2-4ac\), can now be used to determine the number of solutions to the quadratic equation above. If \(\Delta \geq 0\) then there will be at least one point of intersection. \[\therefore (-1)^2-4(1)(c) \geq 0\] \[\therefore 1-4c \geq 0 \] \[\therefore c \leq \frac{1}{4} \] |