Definition: The inverse of a function is the “reverse” of ...



Minds onComplete the drawings as if the line was a mirror line. Do not turn the page! Definition: The inverse of a function is the “reverse” of the original function.Example: The inverse of y=x2 is The inverse may or may not be a function. If the inverse is a function, and the original function is f, then f-1 is the name of the inverse. *Be careful! f-1 is read “f inverse” and the -1 is NOT an exponent! It does not mean 1f.In a table of values, the x and y values are switched for the inverse function.If (a, b)∈f, then (b, a)∈f-1. Alternatively, if fa=b, then f-1b=aAlgebraically, you switch the x and y and then re-solve for yThe graph of an inverse is the reflection of f in the y=x line.Example 1a) Find the inverse of f={(0, 3), (1, 3), (2, 3), (3, 3), (4, 3), (5, 3)}.Invariant points are points that do not change during a transformation. What point(s) are invariant?Is the inverse a function? Explain your answer.State the domain and range of f.State the domain and range of the inverse.Example 2 Algebraic InversesFind the inverse of the linear function fx=3x+2.Is the inverse a function?Graph the function and its inverse.*Note: When you have a word problem and the variables are specified, you do NOT switch the variables, this would change the meaning of the equations.Example 3 Application QuestionThe equation used for converting degrees Celsius to degrees Fahrenheit is F=95C+32. (In function notation this would be F(C)=95C+32.)Find the inverse of this function.When would the inverse be useful?Example 4Restricting the domain to make the inverse a functionFind the inverse of h(x)=x2+5 algebraically.Graph the function and it’s inverse.Is the inverse a function?Restrict the domain of h(x) to make the inverse a function.Example 5a) Graph the function fx=2x+3-5. Then graph its inverse. b) Find the inverse of the function algebraically.c) Restrict the domain of the inverse function to make the function match your graph.Example 6 Algebraically find the inverse of gx=5x-3+6. Is the inverse a function?Example 7Algebraically find the inverse of 2x2+56x+29Example 8a) Find the inverse of y=x3. Graph both y=x3 and its inverse.b) Compare the inverses of y=x2 and y=x3Example 9 Show that for fx=4x+5, that ff-1x=x and that f-1fx=x. Explain why this makes sense.Example 10Graph fx=2x-1+3 and its inverse. Is the inverse a function?Advanced:For example 10, what could you write as the equation of the inverse? Look up piecewise functions.ThinkingTry to make a function that is its own inverse. Think of as many functions as possible.Homework: pg. 138 #1ab, 3, 4bd, 5, 7abc, 11, 12, 15ii, iii, iv, 17ab ................
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