How do you graph the function #f(x)=(x+1)^2-4# and its inverse?

Question

Sophie Adams · Accepted Answer

Please read the explanation.

Given:

Graph the function #f(x)=(x+1)^2-4# and its inverse#color(blue)(f^-1(x)#.

A function and it's inverse will be symmetric around the line #color(red)(y=x#

Switch the position of #color(red)(x# and #color(red)(y# variables to find the inverse of a function.

We have,

#y=f(x)=(x+1)^2-4#

Write the quadratic as #color(blue)(x=(y+1)^2 - 4#, after switching #color(red)(x# and #color(red)(y# positions.

Next, solve #color(blue)(x=(y+1)^2-4#, for #color(red)(y#.

Switch sides and rewrite as

#color(blue)((y+1)^2-4=x#

Ad #color(red)(4# to both sides.

#(y+1)^2 - 4 + 4 = x+4#

#i.e., (y+1)^2 - cancel(4) + cancel(4) = x+4#

#i.e., (y+1)^2 =x+4#

#i.e., (y+1) =+-sqrt(x+4)#

Subtract #color(red)(1# from both the sides of the equation.

#i.e., y+1 - 1 =+-sqrt(x+4)-1#

#i.e., y+cancel(1) - cancel(1) =+-sqrt(x+4)-1#

Hence, we get

#y =f^-1(x)=+-sqrt(x+4)-1#

Our final solutions to the quadratic equation are

#y_1 = sqrt(x+4)-1# and

#y_2 = -sqrt(x+4)-1#

Examine the image of the graph containing #color(red)(f(x)# and #color(red)(f^-1(x)# below:

A graphical display calculator may also be used to draw the graph as shown below:

Hope it helps.

Caleb Alexander · Answer

undefined To graph the function $ f(x) = (x+1)^2 - 4 $ and its inverse:

1. Graph the function $ f(x) = (x+1)^2 - 4 $ by plotting points or using transformations of the basic parabola $ y = x^2 $.

2. To find the inverse of $ f(x) $, swap the roles of $ x $ and $ y $ and solve for $ y $.

3. Let $ y = (x+1)^2 - 4 $.

4. Swap $ x $ and $ y $: $ x = (y+1)^2 - 4 $.

5. Solve for $ y $:  
   $ x = (y+1)^2 - 4 $  
   $ x + 4 = (y+1)^2 $  
   $ \sqrt{x+4} = y + 1 $  
   $ y = \sqrt{x+4} - 1 $

6. This expression represents the inverse function $ f^{-1}(x) $.

7. Graph the inverse function $ f^{-1}(x) = \sqrt{x+4} - 1 $ on the same set of axes as $ f(x) $.

8. Ensure the graphs pass the horizontal line test to verify they are inverses of each other.