Write an equation of the form fx ax b from the given graph

Write an equation of the form f(x) = a^x + b from the given graph. Then compute f(2) f(x) = (Use integers or fractions for any numbers in expression) f(2) = (Type an integer or a simplified fraction.)

Solution

Solution:

f(x) = a^x + b

we can see that in graph

that f(-1) = 0

plug this into f(x)

f(-1) = a^-1+b = 0

b = -1/a

Now agian

f(-2) = 2 = a^-2+b

1/a² + b = 2

plug b = -1/a

1/a² -1/a = 2

(1-a)/a² = 2

2a²+a-1 = 0

a= 1/2 and a =-1

Now if a = 1/2

then b = -2

and if a=-1 then b = 1

chose first case when a= 1/2 and b=-2

f(x) = (1/2)^x -2

lets check this f(-1) = 0

(1/2)^-1 -2 = 0

thus it satisfy given condition

again for f(-2) = 2

2 = (1/2)^-2 -2 =2

thus our one function is f(x) = (1/2)^x -2

Let\'s check for other case when a = -1 then b =1

f(x) = (-1)^x +1

Now plug check for the given points f(-1) = 0

f(-1) = (-1)^-1 +1 = -1+1 = 0

again for check for point f(-2)=2

(-1)^-2 +1 = 2

above condtion is not define ( we can take negative in squreroot )

therefore

our required function is

f(x) = (1/2)^x -2

answer

Now

f(2) = (1/2)² -2 = 1/4-2 = -7/4

Answer