QuestionFind the largest domain for which $f(x)=9-x$ is one-to-one, then provide the inverse function for that domain. Use interval notation.

Studdy Solution

STEP 1

Assumptions1. The given function is $f(x)=9-x$ . We need to find the largest domain over which the function is one-to-one3. We also need to find the inverse of the function restricted to that domain

STEP 2

First, let's understand what it means for a function to be one-to-one. A function is one-to-one if every element of the range corresponds to exactly one element of the domain. In other words, no two different elements in the domain of the function have the same function value.

STEP 3

The function $f(x)=9-x$ is a linear function with a negative slope. This means that as $x$ increases, $f(x)$ decreases. Therefore, for every unique value of $x$ , there is a unique value of $f(x)$ . This implies that the function is one-to-one over its entire domain.

STEP 4

The domain of a linear function is all real numbers, or $(-\infty, \infty)$ .

STEP 5

Now that we have determined the domain over which the function is one-to-one, we can find the inverse of the function. The inverse of a function $f(x)$ is denoted as $f^{-1}(x)$ , and it "undoes" the operation of the function. In other words, if $f(x) = y$ , then $f^{-1}(y) = x$ .

STEP 6

To find the inverse of the function, we first replace $f(x)$ with $y$ .
$y =9 - x$

STEP 7

Next, we swap $x$ and $y$ to get the equation for the inverse function.
$x =9 - y$

STEP 8

Finally, we solve for $y$ to get the equation for the inverse function.
$y = - x$ So, $f^{-1}(x) = - x$ .
The largest domain over which the function $f(x)=-x$ is one-to-one is $(-\infty, \infty)$ , and the inverse of the function restricted to that domain is $f^{-1}(x) = - x$ .

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QuestionFind the largest domain for which $f(x)=9-x$ is one-to-one, then provide the inverse function for that domain. Use interval notation.

Studdy Solution

STEP 1

Assumptions1. The given function is $f(x)=9-x$ . We need to find the largest domain over which the function is one-to-one3. We also need to find the inverse of the function restricted to that domain

STEP 2

First, let's understand what it means for a function to be one-to-one. A function is one-to-one if every element of the range corresponds to exactly one element of the domain. In other words, no two different elements in the domain of the function have the same function value.

STEP 3

The function $f(x)=9-x$ is a linear function with a negative slope. This means that as $x$ increases, $f(x)$ decreases. Therefore, for every unique value of $x$ , there is a unique value of $f(x)$ . This implies that the function is one-to-one over its entire domain.

STEP 4

The domain of a linear function is all real numbers, or $(-\infty, \infty)$ .

STEP 5

Now that we have determined the domain over which the function is one-to-one, we can find the inverse of the function. The inverse of a function $f(x)$ is denoted as $f^{-1}(x)$ , and it "undoes" the operation of the function. In other words, if $f(x) = y$ , then $f^{-1}(y) = x$ .

STEP 6

To find the inverse of the function, we first replace $f(x)$ with $y$ .
$y =9 - x$

STEP 7

Next, we swap $x$ and $y$ to get the equation for the inverse function.
$x =9 - y$

STEP 8

Finally, we solve for $y$ to get the equation for the inverse function.
$y = - x$ So, $f^{-1}(x) = - x$ .
The largest domain over which the function $f(x)=-x$ is one-to-one is $(-\infty, \infty)$ , and the inverse of the function restricted to that domain is $f^{-1}(x) = - x$ .

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QuestionFind the largest domain for which f(x)=9−xf(x)=9-xf(x)=9−x is one-to-one, then provide the inverse function for that domain. Use interval notation.

Studdy Solution

STEP 1

Assumptions1. The given function is f(x)=9−xf(x)=9-xf(x)=9−x . We need to find the largest domain over which the function is one-to-one3. We also need to find the inverse of the function restricted to that domain

STEP 2

First, let's understand what it means for a function to be one-to-one. A function is one-to-one if every element of the range corresponds to exactly one element of the domain. In other words, no two different elements in the domain of the function have the same function value.

STEP 3

The function f(x)=9−xf(x)=9-xf(x)=9−x is a linear function with a negative slope. This means that as xxx increases, f(x)f(x)f(x) decreases. Therefore, for every unique value of xxx, there is a unique value of f(x)f(x)f(x). This implies that the function is one-to-one over its entire domain.

STEP 4

The domain of a linear function is all real numbers, or (−∞,∞)(-\infty, \infty)(−∞,∞).

STEP 5

STEP 6

To find the inverse of the function, we first replace f(x)f(x)f(x) with yyy.y=9−xy =9 - xy=9−x

STEP 7

Next, we swap xxx and yyy to get the equation for the inverse function.x=9−yx =9 - yx=9−y

STEP 8

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QuestionFind the largest domain for which $f(x)=9-x$ is one-to-one, then provide the inverse function for that domain. Use interval notation.

Assumptions1. The given function is $f(x)=9-x$ . We need to find the largest domain over which the function is one-to-one3. We also need to find the inverse of the function restricted to that domain

The function $f(x)=9-x$ is a linear function with a negative slope. This means that as $x$ increases, $f(x)$ decreases. Therefore, for every unique value of $x$ , there is a unique value of $f(x)$ . This implies that the function is one-to-one over its entire domain.

The domain of a linear function is all real numbers, or $(-\infty, \infty)$ .

To find the inverse of the function, we first replace $f(x)$ with $y$ .
$y =9 - x$

Next, we swap $x$ and $y$ to get the equation for the inverse function.
$x =9 - y$