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Functions $f$ and $g$ are defined on $ m{R}$, the set of real numbers - Scottish Highers Maths - Question 6 - 2016

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Question 6

Functions-$f$-and-$g$-are-defined-on-$-m{R}$,-the-set-of-real-numbers-Scottish Highers Maths-Question 6-2016.png

Functions $f$ and $g$ are defined on $ m{R}$, the set of real numbers. The inverse functions $f^{-1}$ and $g^{-1}$ both exist. (a) Given $f(x) = 3x + 5$, find $f^{-... show full transcript

Worked Solution & Example Answer:Functions $f$ and $g$ are defined on $ m{R}$, the set of real numbers - Scottish Highers Maths - Question 6 - 2016

Step 1

Given $f(x) = 3x + 5$, find $f^{-1}(x)$

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Answer

To find the inverse function f−1(x)f^{-1}(x), we start by expressing yy in terms of xx:

  1. Set y=f(x)=3x+5y = f(x) = 3x + 5.

  2. Rearrange the equation to isolate xx:

    y−5=3xy - 5 = 3x

    x = rac{y - 5}{3}

  3. By interchanging xx and yy, we get:

    f^{-1}(x) = rac{x - 5}{3}

Step 2

If $g(2) = 7$, write down the value of $g^{-1}(7)$

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Answer

Since g(2)=7g(2) = 7, this implies that g−1(7)g^{-1}(7) is simply the value of 22, because the inverse function will return the original input:

g−1(7)=2g^{-1}(7) = 2

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