Question

Assume the given function is one-to-one. Find the indicated values:x0123456789f(x)8074265391f(1)=AnswerIf f(x)=3 then x=Answerf^{-1}(0)=AnswerIf f^{-1}(x)=7 then x=?Answer

Answer 1

The values are as follows:

-
`\( f(1) = 9 \)`

- If
`\( f(x) = 3 \)`, then
`\( x = 6 \)`

-
`\( f^(-1)(0) = 9 \)`

- If
`\( f^(-1)(x) = 7 \)`, then
`\( x = 2 \)`

The provided image shows a function
`\( f \)` with a set of inputs (x values) and their corresponding outputs (f(x) values). To answer the questions, we need to use the given function values and apply the concept of a one-to-one function, which, by definition, has a unique output for every unique input and vice versa.

Here are the steps to find the required values:

1.
`\( f(1) \):` Find the output when the input
`\( x \)` is 1.

2. If
`\( f(x) = 3 \), then \( x = \)?`: Find the input that gives an output of 3.

3.
`\( f^(-1)(0) \)`: Find the input that corresponds to an output of 0, which is the inverse function value.

4. If
`\( f^(-1)(x) = 7 \)`, then
`\( x = \)`?: Find the output which, when passed through the inverse function, gives an input of 7.

Let's calculate these step by step.

Here are the detailed calculations for the given function:

1. To find
`\( f(1) \)`, we look at the value of the function when the input
`\( x \)` is 1. The output is 9.

2. To find the input
`\( x \)` such that
`\( f(x) = 3 \)`, we look for the input value that corresponds to the output of 3. This input value is 6.

3. To find
`\( f^(-1)(0) \)`, we look for the input value that corresponds to the output of 0. Since the function is one-to-one, the inverse function
`\( f^(-1) \)` will give us the original input for this output, which is 9.

4. To find
`\( x \)` such that
`\( f^(-1)(x) = 7 \)`, we need to find the output
`\( x \)` which, when passed through the inverse function, gives an input of 7. Since
`\( f(7) = 2 \)`, then
`\( f^(-1)(2) = 7 \)`, so
`\( x \)` is 2.

So the values are as follows:

-
`\( f(1) = 9 \)`

- If
`\( f(x) = 3 \)`, then
`\( x = 6 \)`

-
`\( f^(-1)(0) = 9 \)`

- If
`\( f^(-1)(x) = 7 \)`, then
`\( x = 2 \)`

Answer 2

Functions

The table shows the values of x and y that define a one-to-one function.

We can see for example that for x = 7, f(x) = 3.

We can also see that for f(x) = 1, then x = 9.

With those examples in mind, we can find:

f(1) = 0. We look below the value of x=1 and find the value of f(x) = 0

If f(x) = 3, we have to look to which value of x corresponds the value of f(x) = 3. Since the function is guaranteed to be one-to-one, we can say that x = 7

Now we find f^-1(0). This is similar to the previous part where we are given the value of f(x) and find the corresponding value of x. This value is x = 1, thus

f^-1(0) = 1

We are given f^-1(x) = 7. We are required to find the value of x. This is a tricky question because the inverse function gives us the corresponding value of x, so if we know the inverse function value equals 7, then x = 7