Question

Given that f(x)=2x+1
a) find f(2)
b)find f^-1(x)
c)f^-1(7)

Answer 1

a)
`f(2) = 5`, b)
`f^(-1) (x) = (x-1)/(2)`, c)
`f^(-1) (7) = 3`

Explanation:

a) We evaluate the function at
`x = 2`:

`f(2) = 2\cdot (2) + 1`

`f(2) = 4+1`

`f(2) = 5`

b) First, we determine the inverse of the function by algebraic means:

1)
`y = 2\cdot x + 1` Given

2)
`y +(-1) = 2\cdot x + [1+(-1)]` Compatibility with addition/Associative property

3)
`y + (-1) =2\cdot x` Existence of additive inverse/Modulative property

4)
`2^(-1)\cdot [y+(-1)] = (2\cdot 2^(-1))\cdot x` Compatibility with multiplication/Commutative and associative properties

5)
`[y+(-1)]\cdot 2^(-1) = x` Existence of multiplicative inverse/Modulative and commutative properties

6)
`x = [y+(-1)]\cdot 2^(-1)` Symmetry property of equality

7)
`x = (y-1)/(2)` Definitions of subtraction and division

8)
`f^(-1) (x) = (x-1)/(2)`
`x = f_(-1) (x)`/
`y = x`/Result

c) Now we evaluate the expression obtained on b) at the given number:

`f^(-1) (7) = (7-1)/(2)`

`f^(-1) (7) = 3`