Solved on Oct 29, 2023

Find the inverse of $\sqrt{x-3}$ and determine its domain, including any restrictions from the original function.

STEP 1

Assumptions1. The function is $f(x)=\sqrt{x-3}$ . We need to find the inverse of this function and its domain

STEP 2

To find the inverse of a function, we switch the roles of $x$ and $y$ . So, we write the function as $y=\sqrt{x-}$ and then swap $x$ and $y$ to get $x=\sqrt{y-}$ .
$x=\sqrt{y-}$

STEP 3

Now, we need to solve this equation for $y$ to get the inverse function. To do this, we square both sides of the equation to eliminate the square root.
$(x)^2=(\sqrt{y-3})^2$

STEP 4

implify the equation to get $y$ alone on one side.
$x^2=y-3$

STEP 5

Add3 to both sides to isolate $y$ .
$y=x^2+3$ This is the inverse function, which we can write as $f^{-1}(x)=x^2+3$ .

STEP 6

Now, we need to find the domain of the inverse function. The domain of a function is the set of all possible input values (x-values) that will give real output values (y-values).
For the inverse function $f^{-1}(x)=x^2+3$ , the domain is all real numbers because any real number can be squared and then added to3 to give a real number.
However, we also need to consider any restrictions inherited from the original function $f(x)=\sqrt{x-3}$ .

STEP 7

The original function $f(x)=\sqrt{x-3}$ has a domain restriction. The expression under the square root, $x-3$ , must be greater than or equal to0 because we cannot take the square root of a negative number.
So, $x-3 \geq0$ .

STEP 8

olve this inequality for $x$ to find the domain restriction.
$x \geq3$

STEP 9

This means that the original function $f(x)=\sqrt{x-3}$ is defined for all $x$ greater than or equal to3. This restriction is inherited by the inverse function. So, the domain of the inverse function $f^{-}(x)=x^2+3$ is all $x$ greater than or equal to3.
In conclusion, the inverse function of $f(x)=\sqrt{x-3}$ is $f^{-}(x)=x^2+3$ and its domain is all $x$ greater than or equal to3.

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Find the inverse of $\sqrt{x-3}$ and determine its domain, including any restrictions from the original function.

STEP 1

Assumptions1. The function is $f(x)=\sqrt{x-3}$ . We need to find the inverse of this function and its domain

STEP 2

To find the inverse of a function, we switch the roles of $x$ and $y$ . So, we write the function as $y=\sqrt{x-}$ and then swap $x$ and $y$ to get $x=\sqrt{y-}$ .
$x=\sqrt{y-}$

STEP 3

Now, we need to solve this equation for $y$ to get the inverse function. To do this, we square both sides of the equation to eliminate the square root.
$(x)^2=(\sqrt{y-3})^2$

STEP 4

implify the equation to get $y$ alone on one side.
$x^2=y-3$

STEP 5

Add3 to both sides to isolate $y$ .
$y=x^2+3$ This is the inverse function, which we can write as $f^{-1}(x)=x^2+3$ .

STEP 6

STEP 7

The original function $f(x)=\sqrt{x-3}$ has a domain restriction. The expression under the square root, $x-3$ , must be greater than or equal to0 because we cannot take the square root of a negative number.
So, $x-3 \geq0$ .

STEP 8

olve this inequality for $x$ to find the domain restriction.
$x \geq3$

STEP 9

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Find the inverse of x−3\sqrt{x-3}x−3​ and determine its domain, including any restrictions from the original function.

STEP 1

Assumptions1. The function is f(x)=x−3f(x)=\sqrt{x-3}f(x)=x−3​ . We need to find the inverse of this function and its domain

STEP 2

To find the inverse of a function, we switch the roles of xxx and yyy. So, we write the function as y=x−y=\sqrt{x-}y=x−​ and then swap xxx and yyy to get x=y−x=\sqrt{y-}x=y−​.x=y−x=\sqrt{y-}x=y−​

STEP 3

Now, we need to solve this equation for yyy to get the inverse function. To do this, we square both sides of the equation to eliminate the square root.(x)2=(y−3)2(x)^2=(\sqrt{y-3})^2(x)2=(y−3​)2

STEP 4

implify the equation to get yyy alone on one side.x2=y−3x^2=y-3x2=y−3

STEP 5

Add3 to both sides to isolate yyy.y=x2+3y=x^2+3y=x2+3This is the inverse function, which we can write as f−1(x)=x2+3f^{-1}(x)=x^2+3f−1(x)=x2+3.

STEP 6

STEP 7

The original function f(x)=x−3f(x)=\sqrt{x-3}f(x)=x−3​ has a domain restriction. The expression under the square root, x−3x-3x−3, must be greater than or equal to0 because we cannot take the square root of a negative number.So, x−3≥0x-3 \geq0x−3≥0.

STEP 8

olve this inequality for xxx to find the domain restriction.x≥3x \geq3x≥3

STEP 9

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Find the inverse of $\sqrt{x-3}$ and determine its domain, including any restrictions from the original function.

Assumptions1. The function is $f(x)=\sqrt{x-3}$ . We need to find the inverse of this function and its domain

To find the inverse of a function, we switch the roles of $x$ and $y$ . So, we write the function as $y=\sqrt{x-}$ and then swap $x$ and $y$ to get $x=\sqrt{y-}$ .
$x=\sqrt{y-}$

Now, we need to solve this equation for $y$ to get the inverse function. To do this, we square both sides of the equation to eliminate the square root.
$(x)^2=(\sqrt{y-3})^2$

implify the equation to get $y$ alone on one side.
$x^2=y-3$

Add3 to both sides to isolate $y$ .
$y=x^2+3$ This is the inverse function, which we can write as $f^{-1}(x)=x^2+3$ .

The original function $f(x)=\sqrt{x-3}$ has a domain restriction. The expression under the square root, $x-3$ , must be greater than or equal to0 because we cannot take the square root of a negative number.
So, $x-3 \geq0$ .

olve this inequality for $x$ to find the domain restriction.
$x \geq3$