Question

Can somebody help me with #14 ? I don’t know how to start it off and solve it

Answer 1

Given the function:

`f(x)=(x-4)^2`

• You can graph it.

By definition, this is the graph of the Parent Function (the simplest form) of Quadratic Functions:

The equation of this Parent Function is:

`y=x^2`

You can identify that the function given in the exercise is like the Parent Function graphed above, but translated 4 units to the right. Because, according to the Transformation Rules for Functions, when:

`f(x-h)`

The function is shifted right "h" units.

Therefore, you can graph the function provided in the exercise:

According to the instruction given in the exercise, you have to find the domain on which the function is one-to-one and non-decreasing.

By analyzing the graph, you can determine that the function increases (goes up) on this interval:

`\lbrack4,+\infty)`

In order for that portion (the portion on the right, which is the one increasing) to be one-to-one, it has two passes the Vertical Line Test. This states that if the vertical lines intersect the graph at more than one point, it is not a One-to-One Function.

In this case, you get:

Since all the lines intersect the graph at one point, then it is a One-to-one Function.

By definition, the Domain of a function is the set of x-values for which it is defined.

Therefore, you can determine that the domain on which the function is one-to-one and non-decreasing is:

`Domain:\lbrack4,\infty)`

• In order to find an inverse of the function of this domain, you need to follow these steps:

1. Rewrite the function in this form:

`y=(x-4)^2`

2. Solve for "x":

`\begin{gathered} √(y)=√((x-4)^2) \\ \\ y=x-4 \\ x=√(y)+4 \end{gathered}`

3. Swap the variables:

`y=√(x)+4`

4. Rewrite it as:

`f^(-1)(x)=√(x)+4`

Keeping in mind the definition of Domain, you need to remember that a square root is not defined when its Radicand (the value inside the root) is negative.

Therefore, the Domains are the same:

`Domain:\lbrack4,\infty)`

Hence, the answer is:

• Domain on which the function is one-to-one and non-decreasing:

`Domain:\lbrack4,\infty)`

• Inverse of function on that domain:

`f^(-1)(x)=√(x)+4`