Let y be a general exponential function:
[tex]y=ka^x+b[/tex]For a function to be increasing, its derivative must be positive. Notice that the derivative of this function is:
[tex]y^{\prime}=k\cdot\ln (a)\cdot a^x[/tex]Since a^x is always positive, the sign depents only on the sign of k and ln(a).
For the product k*ln(a) to be positive, their signs must be equal. Notice that ln(a) will be positive whenever a>1, and negative whenever 0k is positive and a>1, or if k is negative and 0.
Notice that the function:
[tex]y=7\cdot(\frac{3}{2})^x[/tex]Satisfies that 7>0 and 3/2 >1. Then, it is an increasing exponential function.
Notice that the equations found in options (3) and (4) are not exponential, and the exponential function in option (1) is decreasing because 4>0 and 0<0.75<1.
Therefore, the equation in option (2) is an increasing exponential function.
