# F(x)=ab^{x} where b must be a positive real number

• MHB
• Avalance789
In summary, an exponential function in mathematics is a function with the form f(x)=ab^{x}, where b is a positive real number. It is important for the exponent base to be positive and real because if b<0, the function would not be defined for all values of x. In the case of b=-1 and x=1/2, the function would involve complex numbers, specifically sqrt{-1}. Therefore, we cannot use it for certain applications. This leads to the introduction of complex numbers in mathematics.
Avalance789
Quote: "In mathematics, an exponential function is a function of the form

f(x)=ab^{x}

where b is a POSITIVE REAL number"

Wait. Give me a reason, why exponent base must be positive and real?

What happens if b<0?

Avalance789 said:
Quote: "In mathematics, an exponential function is a function of the form

f(x)=ab^{x}

where b is a POSITIVE REAL number"

Wait. Give me a reason, why exponent base must be positive and real?

What happens if b<0?
What happens if b = -1 and x = 1/2?

-Dan

It means sqrt{-1}?

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Avalance789 said:
It means sqrt{-1}?

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Exactly. And as i is not real number we can't use it for what you seem to be working with.

-Dan

Ok, got it. So from this point we get into complex numbers.

Thank you!

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## What is the meaning of "F(x)=ab^{x} where b must be a positive real number"?

The function F(x) represents the output or dependent variable, while a and b are constants. The variable x is the input or independent variable. The value of b must be a positive real number, which means it cannot be a negative number or zero.

## What is the significance of b being a positive real number in this function?

The value of b determines the rate of change or growth of the function. A positive value for b indicates that the function is increasing, while a negative value would make it decreasing. A real number is used to make the function continuous and applicable to real-world situations.

## How is the graph of this function affected by changing the value of b?

The graph of F(x)=ab^{x} is an exponential curve, and changing the value of b will affect the steepness of the curve. A larger value of b will result in a steeper curve, while a smaller value will result in a flatter curve. The function will always pass through the point (0,a) on the y-axis.

## Can b be equal to 0 or a negative number in this function?

No, b cannot be 0 or a negative number in this function. This is because raising a number to a negative or zero power will result in a fraction or undefined value, which is not allowed in the domain of a function. Additionally, a negative value for b would make the function decreasing, which goes against the definition of an exponential function.

## What are some real-life applications of this function?

The function F(x)=ab^{x} is commonly used to model exponential growth or decay in various fields such as finance, population growth, and radioactive decay. It can also be used to represent the growth of bacteria, the depreciation of assets, and the spread of diseases. In general, any situation where a quantity increases or decreases at a constant rate can be modeled using this function.

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