Lesson 6 Inverse functions We have already seen some functions that are the inverse of each other. The functions f(x) = x2 and g(x) = √x are the inverse of each other if we limit the x values to f(x) = x2 and g(x) = f(2) = 22 = 4 and g(4) = √4 = 2 f(g(a)) = = a g(f(a)) = = a It doesn't matter which of the two functions f(x) or g(x) we apply first, the result is the same. The range of one fuction becomes the domain of the other. A function has an inverse only if it is one - to - one and onto. In this case we limit our values of x to non negative numbers so that the function f(x) = x2 satisfiesthis condition. In other words a function has an inverse if it is increasing or decreasing on it's domain. The function g(x) = √x is increasing and can only take non - negative numbers which again means that f(x) = x2 is limited to non - negative numbers. We can find the equation of an inverse function algebraically by solving the equation of the function for x. Look at the following example: Example 1 Find the inverse of the following functions: a) f(x) = 2x + 4 y = 2x + 4 put y instead of f(x) �2x = 4 � y x = �2 + �y We have divided through by �2 This is an equation where x is a function of y. The name of the variable doesn't matter so we can exchange the x and the y . If we call this function g we get the equation of the inverse function of f(x). g(x) = y = �x � 2. We can check our result by putting in a number. f(1) = 2�1 + 4 = 6 and g(6) = ��6 � 2 = 3 � 2 = 1 If we use the general value a, we get f(a) = 2a + 4 and g(2a+4) = ��(2a+4) � 2 = a + 2 � 2 = a b) f(x) = sin 2x, Df = [�/4,/4� , f(x) is increasing on this interval and therefore has an inverse function. y = sin 2x 2x = sin�1 y x = � sin�1 y The inverse function is g(x) = � sin�1x. In most books the inverse of a function is written using the index �1 so that f(x) has the inverse functionf �1(x). This does not mean 1/f, it's simply a notation for the inverse function. In the above example f �1(x) = � sin�1x. Check. f(/12) = sin 2/12 = � f�1(�) = ��sin�1� = ��/6 = /12 c) f(x) = e2x y = e2x ln y = ln e2x= 2x x = � ln y The inverse function of f(x) is therefore f�1(x) = � ln x. d) f(x) = x2 � 1, Df = R+ y = x2 � 1 x2 = y + 1
e)
The inverse function is If the function f(x) is either always increasing or always decreasing then it has an inverse function f �1(x). The Range of f(x) becomes the Domain of f �1(x). We find the equation of the inverse by solving the equation y = f(x) for x. Example 2 How do we need to limit the domain of the function f(x) = sin x so that it has an inverse function? We know that a continuous function that is always increasing ( or decreasing ) has an inverse. So we look at the derivative of the function f(x) = sin x , f�(x) = cos x.
Using the unit circle we can see that cos x is positive from �/2 to /2 and negative on the rest of the circle. Look at the graph of f(x) = sin x.
We see that the graph is increasing on �/2 < x
< /2 so f(x) = sin x has an inverse if we limit the domain to this interval. Example 3 Find the interval on which the function f(x) = x2 � 4x + 3 is increasing, limit the domain to this interval and then find the formula for the inverse function. Finally draw the graph of both f(x) and f�1(x) in the same coordinate system. We begin by finding the vertex of the parabola by differentiating and finding where the derivative is zero. ( The tangent to f(x) at the vertex is horizontal and therefore the derivative is zero ) f(x) = x2 � 4x + 3 f�(x) = 2x � 4 = 0 2x = 4 x = 2 The vertex is where x = 2 and the function is increasing after that. We therefore choose the domain
To find the equation of the inverse we need to solve the equation y = x2 � 4x + 3 for x.
Now we draw the graph by first making a table of values. Calculating f(2) tells us that f(x) = y takes values from �1 upwards ( the function is increasing ). We need therefore to start by finding f�1(�1).
The two graphs are shown below. We note that the two graphs are mirror images of each other in the line y = x. (the line that bisects the angle between the x and y axes). We can also see this from the table of values. Each point on f(x) is the mirror image of a point on The graph of a function and it's inverse function are always the mirror image of each other in the line y = x. Example 4 Look at the function f(x) = exand its inverse function g(x) = ln x. The function f(x) = ex can take any value of x so its domain is all the real numbers R. The Range of a function is the same as the Domain of it's inverse and the Domain of a function is the Range of its inverse. The graphs of the two functions are shown below. Notice that they are mirror images of each other in the line y = x . Example 5 Find the interval on which the function is increasing. Choosing this interval as the domain find the equation of its inverse function then draw both graphs in the same coordinate system. Differentiate, using the chain rule, to find where the function is increasing and where decreasing,
The denominator is always positive so its the x in the numerator that tells us where the function is increasing or decreasing. The function is increasing for non negative values of x so we choose the domain
Next we solve for x to find the inverse. y = 1 + (x2 + 1)� y � 1 = (x2 + 1)� (y � 1)2 = x2 + 1 x2 = (y � 1)2�1 = y2 � 2y + 1 � 1 = y2 � 2y
The graphs of f(x), f �1(x) and the line y = x look like this.
Example 6 Find the inverse of the function and draw the graphs in the same coordinate system. The graph has a vertical asymptote x = 1 and a horizontal asymptote y = 2. The domain does not contain x = 1. We differentiate to find the slope of the graph. The denominator is always positive and the numerator always negative which means that the slope of the graph is always negative. So the function is decreasing on all it's domain. We find the inverse function by solving for x.
This function has a vertical asymptote in x = 2 and a horizontal asymptote in y = 1. This is the exact opposite to f(x) which has a vertical asymptote in x = 1 and a horizontal asymptote in y = 2. Look at the two graphs. The graphs and their asymptotes are mirror images of each other in the line y = x. Practise these methods and then take test 6 in functions 2. ps. Remember the check list! What is the amount of increase or decrease in a function usually written as a fraction?The amount of increase or decrease in a function usually written as a fraction. Often considered the slope, this is the comparison of two different quantities that are changing. A set of ordered pairs (x, y). It may or may not also be a function.
What term refers to the set of all Y values in a relation?The range is the set of all y or output values. We may describe it as the collection of the second values in the ordered pairs.
Which function is increasing on the interval (Since, x and y are arbitrary values, therefore, f (x) < f (y) whenever x < y. Therefore, the interval (-∞, ∞) is a strictly increasing interval for f(x) = 3x + 5.
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