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QuestionIf ff is one-to-one and f(2)=7f(2)=7, find f−1(7)f^{-1}(7). Then if f−1(−7)=−9f^{-1}(-7)=-9, find f(−9)f(-9).

Studdy Solution

STEP 1

Assumptions1. The function ff is a one-to-one function. . For a one-to-one function, if f(a)=bf(a) = b, then f−1(b)=af^{-1}(b) = a.
3. Given that f()=7f() =7 and f−1(−7)=−9f^{-1}(-7) = -9.

STEP 2

We can use the property of one-to-one functions to find f−1(7)f^{-1}(7). According to the property, if f(a)=bf(a) = b, then f−1(b)=af^{-1}(b) = a.

STEP 3

Substitute the given values into the property. We know that f(2)=7f(2) =7, so we can say that f−1(7)=2f^{-1}(7) =2.

STEP 4

So, f−1(7)=2f^{-1}(7) =2 is the solution for part (a).

STEP 5

Similarly, we can use the property of one-to-one functions to find f(−9)f(-9). According to the property, if f(a)=bf(a) = b, then f−1(b)=af^{-1}(b) = a.

STEP 6

Substitute the given values into the property. We know that f−1(−)=−9f^{-1}(-) = -9, so we can say that f(−9)=−f(-9) = -.

STEP 7

So, f(−9)=−7f(-9) = -7 is the solution for part (b).
The solutions are f−1(7)=2f^{-1}(7) =2 and f(−9)=−7f(-9) = -7.

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