graph the equation and it's inverse. describe it after.
Answers
Step-by-step explanation:
Inverse functions, in the most general sense, are functions that "reverse" each other. For example, if fff takes aaa to bbb, then the inverse, f^{-1}f
−1
f, start superscript, minus, 1, end superscript, must take bbb to aaa.
Or in other words, f(a)=b \iff f^{-1}(b)=af(a)=b⟺f
−1
(b)=af, left parenthesis, a, right parenthesis, equals, b, \Longleftrightarrow, f, start superscript, minus, 1, end superscript, left parenthesis, b, right parenthesis, equals, a.
Before we start...
In this lesson, we will find the inverse function of f(x)=3x+2f(x)=3x+2f, left parenthesis, x, right parenthesis, equals, 3, x, plus, 2.
Before we do that, let's first think about how we would find f^{-1}(8)f
−1
(8)f, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis.
To find f^{-1}(8)f
−1
(8)f, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis, we need to find the input of fff that corresponds to an output of 888. This is because if f^{-1}(8)=xf
−1
(8)=xf, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis, equals, x, then by definition of inverses, f(x)=8f(x)=8f, left parenthesis, x, right parenthesis, equals, 8.
\begin{aligned} f(x) &= 3 x+2\\\\ 8 &= 3 x+2 &&\small{\gray{\text{Let f(x)=8}}} \\\\6&=3x &&\small{\gray{\text{Subtract 2 from both sides}}}\\\\ 2&=x &&\small{\gray{\text{Divide both sides by 3}}} \end{aligned}
f(x)
8
6
2
=3x+2
=3x+2
=3x
=x
Let f(x)=8
Subtract 2 from both sides
Divide both sides by 3
So f(2)=8f(2)=8f, left parenthesis, 2, right parenthesis, equals, 8 which means that f^{-1}(8)=2f
−1
(8)=2f, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis, equals, 2
Finding inverse functions
We can generalize what we did above to find f^{-1}(y)f
−1
(y)f, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis for any yyy. Why did we use y here?
To find f^{-1}(y)f
−1
(y)f, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis, we can find the input of fff that corresponds to an output of yyy. This is because if f^{-1}(y)=xf
−1
(y)=xf, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis, equals, x then by definition of inverses, f(x)=yf(x)=yf, left parenthesis, x, right parenthesis, equals, y.
\begin{aligned} f(x) &= 3 x+2\\\\ y &= 3 x+2 &&\small{\gray{\text{Let f(x)=y}}} \\\\y-2&=3x &&\small{\gray{\text{Subtract 2 from both sides}}}\\\\ \dfrac{y-2}{3}&=x &&\small{\gray{\text{Divide both sides by 3}}} \end{aligned}
f(x)
y
y−2
3
y−2
=3x+2
=3x+2
=3x
=x
Let f(x)=y
Subtract 2 from both sides
Divide both sides by 3
So f^{-1}(y)=\dfrac{y-2}{3}f
−1
(y)=
3
y−2
f, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis, equals, start fraction, y, minus, 2, divided by, 3, end fraction.
Since the choice of the variable is arbitrary, we can write this as f^{-1}(x)=\dfrac{x-2}{3}f
−1
(x)=
3
x−2
f, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis, equals, start fraction, x, minus, 2, divided by, 3, end fraction.