Video Transcript
The graphs of ๐ of ๐ฅ which is equal to ๐ฅ cubed plus ๐ and its inverse ๐ inverse of ๐ฅ intersect at three points, one of which is four-fifths, four-fifths. Determine the value of ๐. Find the ๐ฅ-coordinate of the point ๐ด marked on the figure. Find the ๐ฅ-coordinate of the point ๐ต marked on the figure.
There are three parts to this question, and we begin by trying to find the value of ๐ in the function ๐ of ๐ฅ is equal to ๐ฅ cubed plus ๐. We are told in the question that the graph of ๐ฆ equals ๐ of ๐ฅ passes through the point four-fifths, four-fifths. We can therefore substitute ๐ฅ equals four-fifths and ๐ of ๐ฅ equals four-fifths into the equation ๐ of ๐ฅ is equal to ๐ฅ cubed plus ๐. We have four-fifths is equal to four-fifths cubed plus ๐. Cubing four-fifths gives us 64 over 125. We can then subtract this from both sides of our equation to calculate the value of ๐. ๐ is equal to four-fifths minus 64 over 125. This is equivalent to 100 over 125 minus 64 over 125. ๐ is therefore equal to 36 over 125. And we have answered the first part of this question.
Next, we recall that the second part of the question asked us to find the ๐ฅ-coordinate of the point ๐ด marked on the figure. Using our previous answer, we know that the figure shows the graph of ๐ of ๐ฅ is equal to ๐ฅ cubed plus 36 over 125 together with its inverse. And since the value of the ๐ฆ-intercept of ๐ of ๐ฅ is 36 over 125 and this is greater than zero, the red plot is ๐ฆ is equal to ๐ of ๐ฅ. And this in turn means that the blue plot is the inverse function.
We can therefore find the coordinates of point ๐ด, which is the ๐ฅ-intercept of the inverse function, by finding the coordinates of the ๐ฆ-intercept of the original function and switching ๐ฅ and ๐ฆ. This is equivalent to reflecting the point across the line ๐ฆ equals ๐ฅ. As already mentioned, the ๐ฆ-intercept of ๐ of ๐ฅ is 36 over 125. So, the point we have labeled ๐ถ has coordinates zero, 36 over 125. Point ๐ด, the ๐ฅ-intercept of the inverse function, therefore has coordinates 36 over 125, zero. And the ๐ฅ-coordinate of the point ๐ด that is required in this part of the question is 36 over 125.
The final part of this question asks us to find the ๐ฅ-coordinate of the point ๐ต that is marked on the figure. This point is not only a point of intersection of the two curves; it also lies on the line ๐ฆ equals ๐ฅ. The point of intersection can therefore be found by solving the system of equations ๐ฆ is equal to ๐ฅ cubed plus 36 over 125 and ๐ฆ is equal to ๐ฅ. We begin by substituting ๐ฆ equals ๐ฅ into the first equation. This gives us ๐ฅ is equal to ๐ฅ cubed plus 36 over 125. We can then multiply through by 125. And subtracting 125๐ฅ from both sides, we have the cubic equation 125๐ฅ cubed minus 125๐ฅ plus 36 equals zero. We know from the figure that one of the points of intersection of the graphs of ๐ and ๐ inverse is ๐ฅ is equal to four-fifths. This means that five ๐ฅ minus four is a factor of 125๐ฅ cubed minus 125๐ฅ plus 36.
In order to find any other factors which will lead us to the points of intersection, we can divide the cubic 125๐ฅ cubed minus 125๐ฅ plus 36 by five ๐ฅ minus four. One way to do this is using the bus stop method of long division as shown. 125๐ฅ cubed minus 125๐ฅ plus 36 is equal to five ๐ฅ minus four multiplied by 25๐ฅ squared plus 20๐ฅ minus nine. We are now in a position where solving the quadratic equation 25๐ฅ squared plus 20๐ฅ minus nine equals zero will give us the other points of intersection.
Applying the quadratic formula where our values of ๐ด, ๐ต, and ๐ถ are 25, 20, and negative nine, respectively, we have ๐ฅ is equal to root 13 minus two divided by five and ๐ฅ is equal to negative root 13 minus two divided by five. We can see from the figure that point ๐ต lies in the first quadrant, so its ๐ฅ-coordinate must be positive. And we can therefore conclude that the ๐ฅ-coordinate of point ๐ต is root 13 minus two divided by five. We have now answered all three parts of the question.
