99.6k views
0 votes
You have been asked to find the inverse off (x) = 3 + V X - 1. What would your first step be?A Square both sides of the equation.Add 1 to both sides of the equation.Subtract 3 from both sides of the equation.Take the square root of both sides.

You have been asked to find the inverse off (x) = 3 + V X - 1. What would your first-example-1

1 Answer

3 votes

We are given a function that is defined by a single variable ( x ) as follows:


f\text{ ( x ) = 3 + }\sqrt{x\text{ - 1}}

We are asked to find an inverse of the given function f ( x ) defined above.

We recall that an inverse of a function is defined as reflection of function f ( x ) across a line defined as:


y=x

Mathematically, we can find the inverse if the function is defined as an equation.

The process of finding an inverse carries two steps:

First step: Make ( x ) the subject of the formula.

Before we start makking ( x ) the subject, we will make the following substitution.


y\text{ = f ( x )}

We replace the above substitution into the function given:


y=\text{ 3 + }\sqrt{x\text{ - 1}}

Now to make ( x ) the subject we will isolate the variable ( x ). We see that the variable ( x ) is accompained by a constant ( 1 ) under the root. We wil first isolate this entire root on the right hand side of the "=" sign.

We do this by subtracting ( 3 ) from both sides of the equation:


\begin{gathered} \text{ y - 3 = 3 +}\sqrt{x\text{ - 1}}\text{ - 3 } \\ \textcolor{#FF7968}{y}\text{\textcolor{#FF7968}{ - 3 = }}\textcolor{#FF7968}{\sqrt{x\text{ - 1}}} \end{gathered}

Then we need to remove the radiacal ( root ) sign from over the head of our subject variable ( x ). We will do this by taking squares on both sides of the equation:


\begin{gathered} (y-3)^2\text{ = \lbrack }√(x-1)\rbrack^2 \\ \textcolor{#FF7968}{(y-3)^2}\text{\textcolor{#FF7968}{ = x - 1}} \end{gathered}

Then we will add ( 1 ) on both sides of the equation to isolate the variable ( x ):


\begin{gathered} (y-3)^2\text{ +1 = x - 1 + 1} \\ \textcolor{#FF7968}{(y-3)^2}\text{\textcolor{#FF7968}{ +1 = x }} \end{gathered}

We have finally made the variable ( x ) the subject of the equation.

Step 2: Substitute y = x

In this step we will make the substitution of line of reflection ( y = x ). We do this by interchanging all ( x ) with ( y ) and all ( y ) with ( x ):


(x-3)^2\text{ + 1 = y}

Then we will express the inverse in standard notation:


y=f^(-1)(\text{ x )}

Therefore,


\textcolor{#FF7968}{f^(-1)(x)=(x-3)^2}\text{\textcolor{#FF7968}{ + 1}}

Answer: The first mathematical operation in finding the inverse was " Subtract ( 3 ) from both sides of the equation "

User Brian Duncan
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories