Question

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

Answer 1

Answer:

a) $f(2) = 5$, b) $f^{-1} (x) = \frac{x-1}{2}$, c) $f^{-1} (7) = 3$

Step-by-step 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) $x = \frac{y-1}{2}$ Definitions of subtraction and division

8) $f^{-1} (x) = \frac{x-1}{2}$ $x = f_{-1} (x)$/$y = x$/Result

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

$f^{-1} (7) = \frac{7-1}{2}$

$f^{-1} (7) = 3$