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` `please explain exponential functions

6 years ago

Dear yazdan,

Exponential functions look somewhat similar to functions you have seen before, in that they involve exponents, but there is a big difference, in that the variable is now the power, rather than the base. Previously, you have dealt with such functions as

f(x) =x^{2}, where the variablexwas the base and the number 2 was the power. In the case of exponentials, however, you will be dealing with functions such asg(x) = 2^{x}, where the base is the fixed number, and the power is the variable.Let's look more closely at the function

g(x) = 2^{x}. To evaluate this function, we operate as usual, picking values ofx, plugging them in, and simplifying for the answers. But to evaluate 2^{x}, we need to remember how exponents work. In particular, we need to remember that negative exponents mean "put the base on the other side of the fraction line".Exponential functions always have some positive number other than 1 as the base. If you think about it, having a negative number (such as –2) as the base wouldn't be very useful, since the even powers would give you positive answers (such as "(–2)

^{2}= 4") and the odd powers would give you negative answers (such as "(–2)^{3}= –8"), and what would you even do with the powers that aren't whole numbers? Also, having 0 or 1 as the base would be kind of dumb, since 0 and 1 to any power are just 0 and 1, respectively; what would be the point? This is why exponentials always have something positive and other than 1 as the base.Please feel free to ask your queries here. We are all IITians and here to help you in your IIT JEE preparation.

All the best.

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Suryakanth –IITB

6 years ago

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