Question

Mackenzie purchased a ticket to the local music festival for $25. The ticket
includes entry and access to up to 3 events at the festival. Each additional
event costs $6. The total cost for attending x events is given by the
functions T(x) and E(x).
T(x) = 6(x-3) + 25, where x>3
E(X) = 25, where 0 sxs3

Answer 1

After a small search, I've found that we want to find the inverses of both functions. We will find that E(x) does not have an inverse function, and the inverse function of T(x) is h(x) = x/6 - 1/6

Here we have a piecewise function:

T(x) = 6(x-3) + 25  if x > 3

E(x) = 25    if 0 ≤ x ≤ 3

First, remember that two functions f(x) and g(x) are inverses if:

f(g(x)) = x

g(f(x)) = x

From that definition we can see that E(x) does not have an inverse, because for any function g(x), we will have:

E(g(x)) = 25

So E(x) can't meet the condition.

Now let's analyze the function T(x)

T(x) = 6(x-3) + 25  

We can rewrite it as:

T(x) = 6x - 6×3 + 25

T(x) = 6x + 1

Note that T(x) is a linear equation, so the inverse will also be a linear equation. Let's assume that the inverse is h(x) = ax + b

We will have:

T(h(x)) = 6×h(x) + 1 = 6(ax + b) + 1  

Now, if these are inverses, we have:

6(ax + b) + 1 = x

6ax + 6b + 1 = x

Then we must have:

6b + 1 = 0

6ax = x

From the first equation, we can get:

6b + 1 = 0

6b = -1

b = -1/6

From the second equation we have:

6ax = x

6a = 1

a = 1/6

Then:

h(x) = x/6 - 1/6

And this is the inverse function of T(x)

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