Answer:
Option B is correct.
The exponential function goes through the points( 1, 16) and (4 , 128) is; \(f(x) = 8(2)^x\)
Step-by-step explanation:
The general form of the exponential function is given by;
\(y=f(x) = ab^x\) where a is the initial value and b is any value greater than 0.
If the exponential function goes through the points (1, 16) and (4, 128) we have;
\(16 = ab^1\)
16 = ab ......[1]
and
\(128 = ab^4\) .....[2]
Divide [2] by [1] we have;
\(\frac{128}{16} =\frac{ab^4}{ab}\)
Simplify:
\(8 = b^3\) or
\(b^3 = 8\)
Taking cube root both sides we get;
\(b = \sqrt[3]{8} =\sqrt[3]{2^3}\)
Simplify:
b = 2
Substitute the value of b = 2 in [1] we get;
\(16 = a \cdot 2\)
Divide both sides by 2 we get;
\(\frac{16}{2} =\frac{2a}{2}\)
Simplify:
a = 8
Substitute the value of a and b in \(f(x) = ab^x\)
then we have;
\(f(x) = 8(2)^x\)
Therefore, the exponential function goes through the points( 1, 16) and (4 , 128) is; \(f(x) = 8(2)^x\)
