Find the inverse of the function y = e5x.On the same diagram sketch the line y = x, and the graphs ofy = e5x and its inverse.(b) Let y = 240 e−7x and set Y = ln y.Find an expression for Y in terms of x and then draw the graph of Yagainst x

To find the inverse of the function y = e^(5x), we first replace y with x and x with y to get x = e^(5y). Then, we solve for y by taking the natural logarithm of both sides to get y = ln(x)/5. This is the inverse of the function.

To sketch the line y = x, simply draw a straight line that passes through the origin (0,0) and has a slope of 1.

To sketch the graph of y = e^(5x), note that this is an exponential function with a base of e and a rate of 5. It will start at the point (0,1) and increase rapidly as x increases.

The inverse function, y = ln(x)/5, will be a reflection of y = e^(5x) over the line y = x. It will start at the point (1,0) and increase slowly as x increases.

For part (b), we have y = 240e^(-7x) and Y = ln(y). Substituting y into the equation for Y gives Y = ln(240e^(-7x)). Using the properties of logarithms, this simplifies to Y = ln(240) - 7x.

To graph Y against x, note that this is a linear function with a y-intercept of ln(240) and a slope of -7. It will start at the point (0, ln(240)) and decrease as x increases.