Math / Algebra

QuestionPart 2 of 4
Use $f(x)$ to complete the table for $f^{-1}(x)$ . $f(x)=x+6$
What is $f^{-1}(-2) ?-8$ What is $f^{-1}(1)$ ?

Studdy Solution

STEP 1

What is this asking? We need to find the inverse function of $f(x) = x + 6$ and use it to calculate specific values. Watch out! Don't confuse $f(x)$ with $f^{-1}(x)$ !
They do opposite things!

STEP 2

1. Find the inverse function
2. Calculate $f^{-1}(-2)$
3. Calculate $f^{-1}(1)$

STEP 3

Alright, let's define the function $f(x) = x + 6$ .
Our mission is to find $f^{-1}(x)$ .
This is the function that "undoes" what $f(x)$ does.
So, if $f(x)$ adds 6, $f^{-1}(x)$ should subtract 6.
Let's see why!

STEP 4

To find the inverse, we start by setting $y = f(x) = x + 6$ .
Now, we want to solve for $x$ in terms of $y$ .
$y = x + 6$

STEP 5

To isolate $x$ , subtract 6 from both sides.
This step "undoes" the addition of 6:
$y - 6 = x$

STEP 6

Now, swap $x$ and $y$ to express the inverse function.
This gives us:
$f^{-1}(x) = x - 6$ Boom! We have our inverse function: $f^{-1}(x) = x - 6$ .

STEP 7

Now, let's calculate $f^{-1}(-2)$ .
Using our inverse function $f^{-1}(x) = x - 6$ , we substitute $-2$ for $x$ :
$f^{-1}(-2) = -2 - 6$

STEP 8

Simplify the expression:
$f^{-1}(-2) = -8$ So, $f^{-1}(-2) = -8$ .
Nice!

STEP 9

Next, let's calculate $f^{-1}(1)$ .
Again, using $f^{-1}(x) = x - 6$ , substitute $1$ for $x$ :
$f^{-1}(1) = 1 - 6$

STEP 10

Simplify the expression:
$f^{-1}(1) = -5$ So, $f^{-1}(1) = -5$ .
Awesome!

STEP 11

The inverse function is $f^{-1}(x) = x - 6$ .
- $f^{-1}(-2) = -8$ - $f^{-1}(1) = -5$

Was this helpful?

QuestionPart 2 of 4
Use $f(x)$ to complete the table for $f^{-1}(x)$ . $f(x)=x+6$
What is $f^{-1}(-2) ?-8$ What is $f^{-1}(1)$ ?

Studdy Solution

STEP 1

What is this asking? We need to find the inverse function of $f(x) = x + 6$ and use it to calculate specific values. Watch out! Don't confuse $f(x)$ with $f^{-1}(x)$ !
They do opposite things!

STEP 2

1. Find the inverse function
2. Calculate $f^{-1}(-2)$
3. Calculate $f^{-1}(1)$

STEP 3

Alright, let's define the function $f(x) = x + 6$ .
Our mission is to find $f^{-1}(x)$ .
This is the function that "undoes" what $f(x)$ does.
So, if $f(x)$ adds 6, $f^{-1}(x)$ should subtract 6.
Let's see why!

STEP 4

To find the inverse, we start by setting $y = f(x) = x + 6$ .
Now, we want to solve for $x$ in terms of $y$ .
$y = x + 6$

STEP 5

To isolate $x$ , subtract 6 from both sides.
This step "undoes" the addition of 6:
$y - 6 = x$

STEP 6

Now, swap $x$ and $y$ to express the inverse function.
This gives us:
$f^{-1}(x) = x - 6$ Boom! We have our inverse function: $f^{-1}(x) = x - 6$ .

STEP 7

Now, let's calculate $f^{-1}(-2)$ .
Using our inverse function $f^{-1}(x) = x - 6$ , we substitute $-2$ for $x$ :
$f^{-1}(-2) = -2 - 6$

STEP 8

Simplify the expression:
$f^{-1}(-2) = -8$ So, $f^{-1}(-2) = -8$ .
Nice!

STEP 9

Next, let's calculate $f^{-1}(1)$ .
Again, using $f^{-1}(x) = x - 6$ , substitute $1$ for $x$ :
$f^{-1}(1) = 1 - 6$

STEP 10

Simplify the expression:
$f^{-1}(1) = -5$ So, $f^{-1}(1) = -5$ .
Awesome!

STEP 11

The inverse function is $f^{-1}(x) = x - 6$ .
- $f^{-1}(-2) = -8$ - $f^{-1}(1) = -5$

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.

QuestionPart 2 of 4Use f(x)f(x)f(x) to complete the table for f−1(x)f^{-1}(x)f−1(x). f(x)=x+6f(x)=x+6f(x)=x+6What is f−1(−2)?−8f^{-1}(-2) ?-8f−1(−2)?−8 What is f−1(1)f^{-1}(1)f−1(1) ?

Studdy Solution

STEP 1

What is this asking? We need to find the inverse function of f(x)=x+6 f(x) = x + 6 f(x)=x+6 and use it to calculate specific values. Watch out! Don't confuse f(x) f(x) f(x) with f−1(x) f^{-1}(x) f−1(x)!They do opposite things!

STEP 2

1. Find the inverse function2. Calculate f−1(−2) f^{-1}(-2) f−1(−2)3. Calculate f−1(1) f^{-1}(1) f−1(1)

STEP 3

Alright, let's **define the function** f(x)=x+6 f(x) = x + 6 f(x)=x+6.Our mission is to find f−1(x) f^{-1}(x) f−1(x).This is the function that "undoes" what f(x) f(x) f(x) does.So, if f(x) f(x) f(x) adds 6, f−1(x) f^{-1}(x) f−1(x) should subtract 6.Let's see why!

STEP 4

To **find the inverse**, we start by setting y=f(x)=x+6 y = f(x) = x + 6 y=f(x)=x+6.Now, we want to solve for x x x in terms of y y y.y=x+6y = x + 6y=x+6

STEP 5

To isolate x x x, **subtract 6 from both sides**.This step "undoes" the addition of 6:y−6=xy - 6 = xy−6=x

STEP 6

Now, **swap x x x and y y y** to express the inverse function.This gives us:f−1(x)=x−6f^{-1}(x) = x - 6f−1(x)=x−6Boom! We have our inverse function: f−1(x)=x−6 f^{-1}(x) = x - 6 f−1(x)=x−6.

STEP 7

Now, let's **calculate** f−1(−2) f^{-1}(-2) f−1(−2).Using our inverse function f−1(x)=x−6 f^{-1}(x) = x - 6 f−1(x)=x−6, we substitute −2-2−2 for x x x:f−1(−2)=−2−6f^{-1}(-2) = -2 - 6f−1(−2)=−2−6

STEP 8

**Simplify** the expression:f−1(−2)=−8f^{-1}(-2) = -8f−1(−2)=−8So, f−1(−2)=−8 f^{-1}(-2) = -8 f−1(−2)=−8.Nice!

STEP 9

Next, let's **calculate** f−1(1) f^{-1}(1) f−1(1).Again, using f−1(x)=x−6 f^{-1}(x) = x - 6 f−1(x)=x−6, substitute 1 1 1 for x x x:f−1(1)=1−6f^{-1}(1) = 1 - 6f−1(1)=1−6

STEP 10

**Simplify** the expression:f−1(1)=−5f^{-1}(1) = -5f−1(1)=−5So, f−1(1)=−5 f^{-1}(1) = -5 f−1(1)=−5.Awesome!

STEP 11

The inverse function is f−1(x)=x−6 f^{-1}(x) = x - 6 f−1(x)=x−6.- f−1(−2)=−8 f^{-1}(-2) = -8 f−1(−2)=−8 - f−1(1)=−5 f^{-1}(1) = -5 f−1(1)=−5

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.

QuestionPart 2 of 4
Use $f(x)$ to complete the table for $f^{-1}(x)$ . $f(x)=x+6$
What is $f^{-1}(-2) ?-8$ What is $f^{-1}(1)$ ?

What is this asking? We need to find the inverse function of $f(x) = x + 6$ and use it to calculate specific values. Watch out! Don't confuse $f(x)$ with $f^{-1}(x)$ !
They do opposite things!

1. Find the inverse function
2. Calculate $f^{-1}(-2)$
3. Calculate $f^{-1}(1)$

Alright, let's define the function $f(x) = x + 6$ .
Our mission is to find $f^{-1}(x)$ .
This is the function that "undoes" what $f(x)$ does.
So, if $f(x)$ adds 6, $f^{-1}(x)$ should subtract 6.
Let's see why!

To find the inverse, we start by setting $y = f(x) = x + 6$ .
Now, we want to solve for $x$ in terms of $y$ .
$y = x + 6$

To isolate $x$ , subtract 6 from both sides.
This step "undoes" the addition of 6:
$y - 6 = x$

Now, swap $x$ and $y$ to express the inverse function.
This gives us:
$f^{-1}(x) = x - 6$ Boom! We have our inverse function: $f^{-1}(x) = x - 6$ .

Now, let's calculate $f^{-1}(-2)$ .
Using our inverse function $f^{-1}(x) = x - 6$ , we substitute $-2$ for $x$ :
$f^{-1}(-2) = -2 - 6$

Simplify the expression:
$f^{-1}(-2) = -8$ So, $f^{-1}(-2) = -8$ .
Nice!

Next, let's calculate $f^{-1}(1)$ .
Again, using $f^{-1}(x) = x - 6$ , substitute $1$ for $x$ :
$f^{-1}(1) = 1 - 6$

Simplify the expression:
$f^{-1}(1) = -5$ So, $f^{-1}(1) = -5$ .
Awesome!

The inverse function is $f^{-1}(x) = x - 6$ .
- $f^{-1}(-2) = -8$ - $f^{-1}(1) = -5$