Chapter 1.

1.1 Problem Statement

{2,4,6}
2*$a
$a*$a

Determine a domain on which f−1 exists, and find a formula for f−1 for this domain of f.

1.2 Step 1

Question Sequence

Question 1.1

The inverse function, f-1(x) exists if and only if f(x)mnIFJEkLQR6TgBpQuRKstlI2e5VMqLeSzswip0RS8d5AJcKro51uGRF5bd6c3Yh8.

Incorrect
Correct

Question 1.2

By looking at the graph of f(x) = x2 - $a2·x, we can see that f(x) zv6snTZ1x/CgMe4nKfH0/A== one-to-one on its entire domain by the x1f8f20FXytk0BktNHiVI7pVlQzXBsa1 line test.

Incorrect
Correct

1.3 Step 2

Since f(x) is not one-to-one, it is not invertible. However, we can restrict the domain to one where f(x) is one-to-one and hence f(x) will be invertible on that domain.

Question Sequence

Question 1.3

Given any parabola, if we restrict the domain to only those numbers to the right (or left) of the vertex, then by the horizontal line test, the function KqDrhW/fkzgmnmA4Ao/tNjgy8Hc= be one-to-one and hence invertible.

2
Incorrect
Correct

Question 1.4

The vertex of the parabola is at x = nc1ItEz0kR4=.

Correct.
Incorrect.

Question 1.5

Thus f(x) will be invertible if it is restricted to the domain

x ≥ nc1ItEz0kR4=

or the domain

x ≤ nc1ItEz0kR4=.

Correct.
Incorrect.

1.4 Step 3

Now that we have a domain on which f(x) is invertible, let's find a formula for f−1(x). To do so, solve y = f(x) for x in terms of y.

There can only be one inverse function and it depends on the restricted domain of f.

Question Sequence

Question 1.6

Solve for x in the restricted domain of f, x ≥ $a.

Solve for x in the restricted domain of f, x ≤ $a.

Where a = nc1ItEz0kR4=.

Correct.
Incorrect.

Question 1.7

Using proper notation for inverse functions, state f−1(x) for the restricted domain x ≥ $a.

Using proper notation for inverse functions, state f−1(x) for the restricted domain x ≤ $a.

Where a = nc1ItEz0kR4=.

Correct.
Incorrect.