Respuesta :
Answer:
The correct option is 2. The given function is an exponential function.
Step-by-step 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
[tex]f(x)=ab^x[/tex]
From the given table it is noticed that the function passing through the points (1,2) and [tex](3,\frac{1}{8})[/tex]. It means the equation must be satisfy by these points.
[tex]2=ab^1[/tex] ..... (1)
[tex]\frac{1}{8}=ab^3[/tex] ..... (2)
Divide equation (2) by equation (1).
[tex]\frac{1}{16}=b^2[/tex]
[tex]\frac{1}{4}=b[/tex]
Put this value in equation (1).
[tex]2=a\frac{1}{4}[/tex]
[tex]a=8[/tex]
The function is
[tex]f(x)=8(\frac{1}{4})^x[/tex]
Check the above equation by remaining points.
[tex]f(-5)=8(\frac{1}{4})^{-5}=8\times 1024=8192[/tex]
[tex]f(-3)=8(\frac{1}{4})^{-3}=8\times 64=512[/tex]
[tex]f(-1)=8(\frac{1}{4})^{-1}=8\times 4=32[/tex]
Since all the point satisfy the function, therefore the given function is an exponential function. Option 2 is correct.
Answer:
Exponential
Step-by-step 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,
[tex]\frac{ 512-8192}{-3+5}\neq \frac{32-512}{-1+32}\neq \frac{2-32}{1+1}\neq \frac{18-2}{3-1}[/tex]
Thus, the function can not be linear,
Now, let us consider the given function f is exponential,
That is,
[tex]f(x)=ab^x[/tex]
By the table, for x = 1, f(x) = 2,
[tex]\implies 2 = ab[/tex]
Also, for x = 3, f(x) = 1/8,
[tex]\implies \frac{1}{8}= ab^3\implies \frac{1}{8} = (2)b^2\implies \frac{1}{16} = b^2\implies b=\frac{1}{4}[/tex]
[tex]\implies a = 8[/tex]
Thus, the exponential function would be,
[tex]f(x)=8(\frac{1}{4})^x[/tex] -----(1)
Now,
[tex]f(-5)=8(\frac{1}{4})^{-5}=8192[/tex]
[tex]f(-3)=8(\frac{1}{4})^{-3}=512[/tex]
[tex]f(-1)=8(\frac{1}{4})^{-1}=32[/tex]
Since, all the given points of the table satisfying the exponential function (1),
Hence, function f is exponential.
