Math / Algebra

QuestionComplete the sentence below. If $f$ is a one-to-one function and $f(9)=8$ , then $f^{-1}(8)=$ $\qquad$ .
If $f$ is a one-to-one function and $f(9)=8$ , then $f^{-1}(8)=$ $\square$ 8. 9. $\frac{1}{8}$ .

Studdy Solution

STEP 1

1. The function $f$ is one-to-one.
2. We are given that $f(9) = 8$ .
3. We need to find $f^{-1}(8)$ .

STEP 2

1. Understand the definition of a one-to-one function.
2. Use the property of inverse functions.
3. Determine the value of $f^{-1}(8)$ .

STEP 3

Understand that a one-to-one function means each output is mapped from a unique input. Therefore, if $f(a) = b$ , then $f^{-1}(b) = a$ .

STEP 4

Use the property of inverse functions: If $f(a) = b$ , then $f^{-1}(b) = a$ .

STEP 5

Given $f(9) = 8$ , apply the inverse function property:
Since $f(9) = 8$ , it follows that $f^{-1}(8) = 9$ .
The value of $f^{-1}(8)$ is:
$\boxed{9}$

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QuestionComplete the sentence below. If $f$ is a one-to-one function and $f(9)=8$ , then $f^{-1}(8)=$ $\qquad$ .
If $f$ is a one-to-one function and $f(9)=8$ , then $f^{-1}(8)=$ $\square$ 8. 9. $\frac{1}{8}$ .

Studdy Solution

STEP 1

1. The function $f$ is one-to-one.
2. We are given that $f(9) = 8$ .
3. We need to find $f^{-1}(8)$ .

STEP 2

1. Understand the definition of a one-to-one function.
2. Use the property of inverse functions.
3. Determine the value of $f^{-1}(8)$ .

STEP 3

Understand that a one-to-one function means each output is mapped from a unique input. Therefore, if $f(a) = b$ , then $f^{-1}(b) = a$ .

STEP 4

Use the property of inverse functions: If $f(a) = b$ , then $f^{-1}(b) = a$ .

STEP 5

Given $f(9) = 8$ , apply the inverse function property:
Since $f(9) = 8$ , it follows that $f^{-1}(8) = 9$ .
The value of $f^{-1}(8)$ is:
$\boxed{9}$

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QuestionComplete the sentence below. If fff is a one-to-one function and f(9)=8f(9)=8f(9)=8, then f−1(8)=f^{-1}(8)=f−1(8)= \qquad .If fff is a one-to-one function and f(9)=8f(9)=8f(9)=8, then f−1(8)=f^{-1}(8)=f−1(8)= □\square□ 8. 9. 18\frac{1}{8}81​.

Studdy Solution

STEP 1

1. The function f f f is one-to-one.2. We are given that f(9)=8 f(9) = 8 f(9)=8.3. We need to find f−1(8) f^{-1}(8) f−1(8).

STEP 2

1. Understand the definition of a one-to-one function.2. Use the property of inverse functions.3. Determine the value of f−1(8) f^{-1}(8) f−1(8).

STEP 3

Understand that a one-to-one function means each output is mapped from a unique input. Therefore, if f(a)=b f(a) = b f(a)=b, then f−1(b)=a f^{-1}(b) = a f−1(b)=a.

STEP 4

Use the property of inverse functions: If f(a)=b f(a) = b f(a)=b, then f−1(b)=a f^{-1}(b) = a f−1(b)=a.

STEP 5

Given f(9)=8 f(9) = 8 f(9)=8, apply the inverse function property:Since f(9)=8 f(9) = 8 f(9)=8, it follows that f−1(8)=9 f^{-1}(8) = 9 f−1(8)=9. The value of f−1(8) f^{-1}(8) f−1(8) is:9 \boxed{9} 9​

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QuestionComplete the sentence below. If $f$ is a one-to-one function and $f(9)=8$ , then $f^{-1}(8)=$ $\qquad$ .
If $f$ is a one-to-one function and $f(9)=8$ , then $f^{-1}(8)=$ $\square$ 8. 9. $\frac{1}{8}$ .

1. The function $f$ is one-to-one.
2. We are given that $f(9) = 8$ .
3. We need to find $f^{-1}(8)$ .

1. Understand the definition of a one-to-one function.
2. Use the property of inverse functions.
3. Determine the value of $f^{-1}(8)$ .

Understand that a one-to-one function means each output is mapped from a unique input. Therefore, if $f(a) = b$ , then $f^{-1}(b) = a$ .

Use the property of inverse functions: If $f(a) = b$ , then $f^{-1}(b) = a$ .

Given $f(9) = 8$ , apply the inverse function property:
Since $f(9) = 8$ , it follows that $f^{-1}(8) = 9$ .
The value of $f^{-1}(8)$ is:
$\boxed{9}$