Question

Function f is shown in the table.

What type of function is function f?



linear

exponential

neither linear nor exponential



The table is:
x f(x)
−5 8192
​−3​ 512
​−1​ 32
1 2
3 18

Answer 1

Exponential

Explanation:

Here, the given table,

x −5 −3​ ​−1​ 1 3

f(x) 8192 512 32 2 18,

we know that, a linear function changes at a constant rate,

Here, the rate of change is not constant,

Because,

`( 512-8192)/(-3+5)\\eq (32-512)/(-1+32)\\eq (2-32)/(1+1)\\eq (18-2)/(3-1)`

Thus, the function can not be linear,

Now, let us consider the given function f is exponential,

That is,

`f(x)=ab^x`

By the table, for x = 1, f(x) = 2,

`\implies 2 = ab`

Also, for x = 3, f(x) = 1/8,

`\implies (1)/(8)= ab^3\implies (1)/(8) = (2)b^2\implies (1)/(16) = b^2\implies b=(1)/(4)`

`\implies a = 8`

Thus, the exponential function would be,

`f(x)=8((1)/(4))^x` -----(1)

Now,

`f(-5)=8((1)/(4))^(-5)=8192`

`f(-3)=8((1)/(4))^(-3)=512`

`f(-1)=8((1)/(4))^(-1)=32`

Since, all the given points of the table satisfying the exponential function (1),

Hence, function f is exponential.

Answer 2

The correct option is 2. The given function is an exponential function.

Explanation:

From the given table it is noticed that the rate of change is not constant, therefore it is not a linear function.

The general form of an exponential function is

`f(x)=ab^x`

From the given table it is noticed that the function passing through the points (1,2) and
`(3,(1)/(8))`. It means the equation must be satisfy by these points.

`2=ab^1` ..... (1)

`(1)/(8)=ab^3` ..... (2)

Divide equation (2) by equation (1).

`(1)/(16)=b^2`

`(1)/(4)=b`

Put this value in equation (1).

`2=a(1)/(4)`

`a=8`

The function is

`f(x)=8((1)/(4))^x`

Check the above equation by remaining points.

`f(-5)=8((1)/(4))^(-5)=8* 1024=8192`

`f(-3)=8((1)/(4))^(-3)=8* 64=512`

`f(-1)=8((1)/(4))^(-1)=8* 4=32`

Since all the point satisfy the function, therefore the given function is an exponential function. Option 2 is correct.