Math / Geometry

Questionmplete the following sentence. ${ }^{-1}$ denotes the inverse of a function $f$ , then the graphs of $f$ and $f^{-1}$ are symmetric with respect to the line $\square$ . (pe an equation.)

Studdy Solution

STEP 1

What is this asking? We need to find the line that acts as a mirror between the graph of a function and the graph of its inverse. Watch out! Don't mix up inverse functions with reciprocal functions!
The inverse is not 1 divided by the function.

STEP 2

1. Understand Inverse Functions
2. Visualize the Symmetry
3. Find the Line of Symmetry

STEP 3

Let's remember what an inverse function is!
If we have a function $f(x)$ that takes an input $x$ and gives us an output $y$ , then the inverse function, written as $f^{-1}(y)$ , takes that output $y$ and gives us back the original input $x$ .
It reverses the process!

STEP 4

Think of it like a gift exchange. $f(x)$ is like wrapping the gift $x$ . $f^{-1}(y)$ is like unwrapping the gift $y$ to get back the original gift $x$ .

STEP 5

Imagine plotting a few points on the graph of $f(x)$ .
Let's say one point is $(a, b)$ , which means $f(a) = b$ .
Now, because $f^{-1}$ does the opposite, we know that $f^{-1}(b) = a$ .
This means the point $(b, a)$ is on the graph of $f^{-1}(y)$ .

STEP 6

Let's take another point on $f(x)$ , say $(c, d)$ .
Then $f(c) = d$ , and so $f^{-1}(d) = c$ .
This gives us the point $(d, c)$ on the graph of $f^{-1}(y)$ .
See the pattern?
We're just swapping the x and y coordinates!

STEP 7

Now, think about what line would act as a mirror to reflect the point $(a, b)$ to $(b, a)$ .
Or $(c, d)$ to $(d, c)$ .
It's the line where the x-coordinate and the y-coordinate are equal!

STEP 8

That line is simply $y = x$ !
If you fold the graph paper along the line $y = x$ , the graphs of $f(x)$ and $f^{-1}(x)$ will lie perfectly on top of each other.

STEP 9

The graphs of $f$ and $f^{-1}$ are symmetric with respect to the line $y = x$ .

Was this helpful?

Questionmplete the following sentence. ${ }^{-1}$ denotes the inverse of a function $f$ , then the graphs of $f$ and $f^{-1}$ are symmetric with respect to the line $\square$ . (pe an equation.)

Studdy Solution

STEP 1

What is this asking? We need to find the line that acts as a mirror between the graph of a function and the graph of its inverse. Watch out! Don't mix up inverse functions with reciprocal functions!
The inverse is not 1 divided by the function.

STEP 2

1. Understand Inverse Functions
2. Visualize the Symmetry
3. Find the Line of Symmetry

STEP 3

Let's remember what an inverse function is!
If we have a function $f(x)$ that takes an input $x$ and gives us an output $y$ , then the inverse function, written as $f^{-1}(y)$ , takes that output $y$ and gives us back the original input $x$ .
It reverses the process!

STEP 4

Think of it like a gift exchange. $f(x)$ is like wrapping the gift $x$ . $f^{-1}(y)$ is like unwrapping the gift $y$ to get back the original gift $x$ .

STEP 5

Imagine plotting a few points on the graph of $f(x)$ .
Let's say one point is $(a, b)$ , which means $f(a) = b$ .
Now, because $f^{-1}$ does the opposite, we know that $f^{-1}(b) = a$ .
This means the point $(b, a)$ is on the graph of $f^{-1}(y)$ .

STEP 6

Let's take another point on $f(x)$ , say $(c, d)$ .
Then $f(c) = d$ , and so $f^{-1}(d) = c$ .
This gives us the point $(d, c)$ on the graph of $f^{-1}(y)$ .
See the pattern?
We're just swapping the x and y coordinates!

STEP 7

Now, think about what line would act as a mirror to reflect the point $(a, b)$ to $(b, a)$ .
Or $(c, d)$ to $(d, c)$ .
It's the line where the x-coordinate and the y-coordinate are equal!

STEP 8

That line is simply $y = x$ !
If you fold the graph paper along the line $y = x$ , the graphs of $f(x)$ and $f^{-1}(x)$ will lie perfectly on top of each other.

STEP 9

The graphs of $f$ and $f^{-1}$ are symmetric with respect to the line $y = x$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Questionmplete the following sentence. −1{ }^{-1}−1 denotes the inverse of a function fff, then the graphs of fff and f−1f^{-1}f−1 are symmetric with respect to the line □\square□ . (pe an equation.)

Studdy Solution

STEP 1

What is this asking? We need to find the line that acts as a mirror between the graph of a function and the graph of its inverse. Watch out! Don't mix up inverse functions with reciprocal functions!The inverse is *not* 1 divided by the function.

STEP 2

1. Understand Inverse Functions2. Visualize the Symmetry3. Find the Line of Symmetry

STEP 3

Let's remember what an **inverse function** is!If we have a function f(x)f(x)f(x) that takes an input xxx and gives us an output yyy, then the **inverse function**, written as f−1(y)f^{-1}(y)f−1(y), takes that output yyy and gives us back the original input xxx.It *reverses* the process!

STEP 4

Think of it like a gift exchange. f(x)f(x)f(x) is like wrapping the gift xxx. f−1(y)f^{-1}(y)f−1(y) is like unwrapping the gift yyy to get back the original gift xxx.

STEP 5

STEP 6

Let's take another point on f(x)f(x)f(x), say (c,d)(c, d)(c,d).Then f(c)=df(c) = df(c)=d, and so f−1(d)=cf^{-1}(d) = cf−1(d)=c.This gives us the point (d,c)(d, c)(d,c) on the graph of f−1(y)f^{-1}(y)f−1(y).See the pattern?We're just swapping the x and y coordinates!

STEP 7

Now, think about what line would act as a mirror to reflect the point (a,b)(a, b)(a,b) to (b,a)(b, a)(b,a).Or (c,d)(c, d)(c,d) to (d,c)(d, c)(d,c).It's the line where the x-coordinate and the y-coordinate are equal!

STEP 8

That line is simply y=xy = xy=x!If you fold the graph paper along the line y=xy = xy=x, the graphs of f(x)f(x)f(x) and f−1(x)f^{-1}(x)f−1(x) will lie perfectly on top of each other.

STEP 9

The graphs of fff and f−1f^{-1}f−1 are symmetric with respect to the line y=xy = xy=x.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Questionmplete the following sentence. ${ }^{-1}$ denotes the inverse of a function $f$ , then the graphs of $f$ and $f^{-1}$ are symmetric with respect to the line $\square$ . (pe an equation.)

What is this asking? We need to find the line that acts as a mirror between the graph of a function and the graph of its inverse. Watch out! Don't mix up inverse functions with reciprocal functions!
The inverse is not 1 divided by the function.

1. Understand Inverse Functions
2. Visualize the Symmetry
3. Find the Line of Symmetry

Let's remember what an inverse function is!
If we have a function $f(x)$ that takes an input $x$ and gives us an output $y$ , then the inverse function, written as $f^{-1}(y)$ , takes that output $y$ and gives us back the original input $x$ .
It reverses the process!

Think of it like a gift exchange. $f(x)$ is like wrapping the gift $x$ . $f^{-1}(y)$ is like unwrapping the gift $y$ to get back the original gift $x$ .

Let's take another point on $f(x)$ , say $(c, d)$ .
Then $f(c) = d$ , and so $f^{-1}(d) = c$ .
This gives us the point $(d, c)$ on the graph of $f^{-1}(y)$ .
See the pattern?
We're just swapping the x and y coordinates!

Now, think about what line would act as a mirror to reflect the point $(a, b)$ to $(b, a)$ .
Or $(c, d)$ to $(d, c)$ .
It's the line where the x-coordinate and the y-coordinate are equal!

That line is simply $y = x$ !
If you fold the graph paper along the line $y = x$ , the graphs of $f(x)$ and $f^{-1}(x)$ will lie perfectly on top of each other.

The graphs of $f$ and $f^{-1}$ are symmetric with respect to the line $y = x$ .