How do you find the inverse of #f(x)= 1/3x + 2# and is it a function?

Question

How do you find the inverse of  #f(x)= 1/3x + 2# and is it a function?

James Allen · Accepted Answer

#color(blue)(g(x)=3x-6)# is a function
(which is the inverse of #f(x)=1/3x+2#)

If #g(x)# is the inverse of #f(x)# then by definition #color(white)("XXX")g(f(x))=x# and #color(white)("XXX")f(g(x))=x# Since #f(color(red)(x))=1/3color(red)(x)+2# then #color(white)("XXX")f(color(red)(g(x)))=1/3color(red)(g(x))+2# But from our original definition we know that #color(white)("XXX")f(g(x))=x# Therefore #color(white)("XXX")f(g(x))=1/3g(x)+2=x# #color(white)("XXX")rArr1/3g(x)=x-2# #color(white)("XXX")rArrg(x)=3x-6# Since any value of #x# gives a single value for #g(x)# it follows that #g(x)# is a function.

Anna Allen · Answer

undefined To find the inverse of a function, follow these steps:

1. Replace f(x) with y.
2. Swap x and y to interchange the dependent and independent variables.
3. Solve the equation obtained for y in terms of x.
4. Replace y with the notation for the inverse function, usually denoted as f^(-1)(x).

For the function $f(x) = \frac{1}{3}x + 2$, let's find its inverse:

1. Replace $f(x)$ with $y$: 
   $y = \frac{1}{3}x + 2$

2. Swap x and y: 
   $x = \frac{1}{3}y + 2$

3. Solve for y: 
   $x - 2 = \frac{1}{3}y$
   $3(x - 2) = y$
   $y = 3x - 6$

4. Replace y with $f^(-1)(x)$: 
   $f^(-1)(x) = 3x - 6$

Now, to determine if the inverse is also a function, we need to check if each input value (x) has exactly one output value (y). Since the inverse function $f^(-1)(x) = 3x - 6$ passes the vertical line test, where each vertical line crosses the graph at most once, it is indeed a function.