Consider f(x) = x2 + 3, x < 0. What is the inverse of this function?

Question

1. If f(x) = 2x + 1 and g(x) = x², what is f o g? What is g o f?

2. Consider f(x) = 2x + 3. Find the inverse of this function. Verbally, compare what f does to x (“multiply by 2, then add 3”) to what f^-1 does to x.

3. Consider f(x) = x² + 3, x < 0. What is the inverse of this function?

Answer 1

1) To do the composition f ο g, you plug g(x) everywhere you see an x in f(x).  f(g(x)) = 2(x²) + 1 2x² + 1

To find the g ο f, plug f(x) everywhere you see an x in g(x).  g(f(x)) = (2x+1)² = 4x² + 4x + 1

 

2) To find the inverse, you switch x and y and solve for y:

 

y = 2x + 3

x = 2y + 3 (switch x and y)

x - 3 = 2y (subtract 3 from both sides)

y = (1/2)x - (3/2) (divide both sides by 2)

 

3) To find the inverse, you again switch x and y and solve for y:

y = x² + 3

x = y² + 3 (switch x and y)

x - 3 = y² (subtract 3 from both sides)

√(x-3) = y (square root both sides)