Given the function

Question

Given the function f(x) = 0.7e^1.4x

- find its inverse

- state the domain and range of each function

- graph both functions on the same set of axes

Answer 1

f(x) = (7/10)e^(7x/5), the domain is all real numbers; the range is all numbers y > 0.

When you graph that function, think of the parent function y = e^x (passing through (0,1), and (1,e) with a horizontal asymptote y = 0). The transformations for the given f(x) do not shift the horizontal asymptote (because there is no vertical shift). The 7/5 in the exponent is "next to" the x, and causes a horizontal shrink by a factor of 5/7, i.e. for each y value, the x-value of the parent is multiplied by 5/7. Note this does not affect the point (0,1). The 7/10 coefficient multiplies the y-values of the parent by a factor of 7/10, so a vertical shrink by that factor. That means (0,1) is shifted to (0, 0.7).

This graph depicts the horizontal shrink and also displays the final function.

desmos.com/calculator/cgzjpbqeli

To find the inverse function, interchange x and y, then solve for y:

x = (7/10)e^7y/5

e^7y/5 = 10x/7

ln(e^7y/5) = ln(10x/7); if numbers are equal, their natural logs are equal.

(7y/5) ln(e) = ln(10x/7)

y = (5/7)ln(10x/y) ; since ln(e) = 1

g(x) = (5/7)ln(10x/7) ; this is the inverse function.

The inverse of a function interchanges x and y. this means the domain and range are interchanged also.

The domain of the inverse is x > 0. The range is all real numbers.

To graph this function focus on the parent function y = ln(x). This function has a vertical asymptote at x = 0 and passes through the point (1,0). The 10/7 next to the x causes a horizontal shrink by a factor of 7/10. The coefficient 5/7 causes a vertical shrink (change to y-value of the parent) by a factor of 5/7.

desmos.com/calculator/yihfq8ihn9

If you graph both functions on the same set of coordinate axes, they are reflections of each other in the line y = x. The segment joining a point of f(x) with its image on g(x), the inverse, has its midpoint on that line, and is perpendicular to that line. Or, the line y=x is the perpendicular bisector of the segment joining a point of f with its corresponding point on g.

Use the slider on the following graph to see a point on f, its image on g, and the midpoint of the segment joining those points in fact on the line y = x.

desmos.com/calculator/11afkk9cxh