SOLUTION: The function 𝑓(π‘₯) = 10^0.1π‘₯ + 15 is given. Define its domain and range. (5 marks) (b) Determine the analytic expression of its inverse function f -1(x) and clearly

Algebra ->  Linear-equations -> SOLUTION: The function 𝑓(π‘₯) = 10^0.1π‘₯ + 15 is given. Define its domain and range. (5 marks) (b) Determine the analytic expression of its inverse function f -1(x) and clearly      Log On


   

Question 1205584: The function 𝑓(π‘₯) = 10^0.1π‘₯ + 15 is given.
Define its domain and range. (5 marks)

(b) Determine the analytic expression of its inverse function f -1(x) and clearly demonstrate your working. Identify the domain of f -1(x). (10 marks)

(c) Use Excel to create a table of x-y values to evaluate the output of f(x) and create a scatter plot of the function. You may use the following values for x: -2, -1, 0, ΒΌ, 1/3, Β½, 2/3, ΒΎ, 1, 2, 3, 4. Include the table and the graph in your report. (10 marks)
(d) Use Desmos to create a graph for both f(x) and f -1(x) and include it in your report. Comment on the graph. What is the relationship between f(x) and f -1(x)?
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
The function 𝑓(π‘₯) = 10^0.1π‘₯ + 15 is given.
Define its domain and range.
I assume you mean f(x) = 10^(0.1x) + 15 or f%28x%29%22%22=%22%2210%5E%280.1x%29

graph%28400%2C400%2C-12%2C12%2C-12%2C12%2C10%5E%280.1x%29%29

Its domain is %28matrix%281%2C3%2C-infinity%2C%22%2C%22%2Cinfinity%29%29 and its range is %28matrix%281%2C3%2C0%2C%22%2C%22%2Cinfinity%29%29

f%28x%29%22%22=%22%2210%5E%280.1x%29

Write y for f(x)

y%22%22=%22%2210%5E%280.1x%29

Interchange x and y

log%28%28x%29%29%22%22=%22%22log%28%2810%5E%280.1y%29%29%29

Take logs base 10 of both sides:

log%28%28x%29%29%22%22=%22%220.1y

Multiply both sides by 10

10%2Alog%28%28x%29%29%22%22=%22%22y

Swap sides:

y%22%22=%22%2210%2Alog%28%28x%29%29

Replace y by f-1(x)

f%5E%28-1%29%28x%29%22%22=%22%2210%2Alog%28%28x%29%29

Its domain and range of the inverse function f-1(x) are
respectively, the range and domain of the original function f(x).  

That's the green graph below, the inverse is the reflection of the graph
in (or across) the line y = x, the blue dotted line:



Sorry, I'm old-fashioned.  I don't have Excel or Desmos.

Edwin