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

Greetings! Lets solve this shall we ?

So, we must find the inverse, domain and range of the inverse, and domain and range of the original function.

So, our original function is f(x) = 0.7e^1.4x . Lets find the inverse first! We switch the x and y such that

x = 0.7e^1.4(y-1), where y^-1 is the inverse. Then we divide 0.7 from both sides such that

(x / 0.7) = e^1.4(y-1)..........Now we natural log both sides to get

ln(x / 0.7) = lne^1.4(y-1)...........Then we bring the exponent down in front of natural log to get

ln(x / 0.7) = (1.4y^-1)lne.........Note: lne = 1 which is the inverse of e¹ = e: converting from exponential form to logarithmic form! Finally we divide 1.4 to both sides to obtain

y^-1 = [ln(x / 0.7)] / 1.4, where the domain is (0, ∞) and the range is (-∞, ∞). (You can verify by graphing it).

The original function y = 0.7e^1.4x has a domain of (-∞, ∞) and a range of (0,∞). (You can verify this by graphing it)

***Notice that the domain of the original function is the range of the inverse function and the range of the original function is the domain of the inverse function. This is a rule by definition of inverses, majority of the time.

I hope this helped!

Answer 2

Greetings! Lets solve this shall we ?

So, we must find the inverse, domain and range of the inverse, and domain and range of the original function.

So, our original function is f(x) = 0.7e^1.4x . Lets find the inverse first! We switch the x and y such that

x = 0.7e^1.4(y-1), where y^-1 is the inverse. Then we divide 0.7 from both sides such that

(x / 0.7) = e^1.4(y-1)..........Now we natural log both sides to get

ln(x / 0.7) = lne^1.4(y-1)...........Then we bring the exponent down in front of natural log to get

ln(x / 0.7) = (1.4y^-1)lne.........Note: lne = 1 which is the inverse of e¹ = e: converting from exponential form to logarithmic form! Finally we divide 1.4 to both sides to obtain

y^-1 = [ln(x / 0.7)] / 1.4, where the domain is (0, ∞) and the range is (-∞, ∞). (You can verify by graphing it).

The original function y = 0.7e^1.4x has a domain of (-∞, ∞) and a range of (0,∞). (You can verify this by graphing it)

***Notice that the domain of the original function is the range of the inverse function and the range of the original function is the domain of the inverse function. This is a rule by definition of inverses, majority of the time.

I hope this helped!