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What happened if the base of exponential function is less than zero?
If the base of an exponential function is less than zero, the function can exhibit complex behavior. Specifically, if the base is a negative number, the function will not be defined for all real numbers, as it will yield complex numbers for non-integer exponents. Consequently, the exponential function may oscillate between positive and negative values, depending on the exponent's parity, which complicates its interpretation in real-world applications. Thus, exponential functions are typically defined with a positive base for meaningful real-valued outputs.
Why does b have to be greater than 0 in an exponential function?
In an exponential function of the form ( f(x) = b^x ), where ( b ) is the base, ( b ) must be greater than 0 to ensure that the function is defined for all real numbers ( x ). If ( b ) were less than or equal to 0, the function would either be undefined (as in the case of negative bases for non-integer exponents) or not exhibit the characteristic growth behavior of exponential functions. Additionally, a positive base guarantees that the function remains continuous and either increases (for ( b > 1 )) or decreases (for ( 0 < b < 1 )), maintaining its fundamental properties.
On a graph what shape doesn't exponential function make?
An exponential function does not create a linear shape on a graph. Instead, it produces a curve that either rises or falls rapidly, depending on whether the base of the exponent is greater than or less than one. The graph is characterized by its continuous and smooth nature, exhibiting either exponential growth or decay. Additionally, it does not form any circular or parabolic shapes, which are seen in other types of functions.
Find a number less than 534000 and a number greater than 534000 that can be expressed in exponential form?
534,000 to the first exponent
Why do scientists prefer to express numbers in exponential notation and metric units?
Because exponential or scientific notation needs less digits for very large numbers and metric units are preferred because they are in multiples of 10 thus avoiding mistaken calculations.
What happened if the base of exponential function is less than zero?
If the base of an exponential function is less than zero, the function can exhibit complex behavior. Specifically, if the base is a negative number, the function will not be defined for all real numbers, as it will yield complex numbers for non-integer exponents. Consequently, the exponential function may oscillate between positive and negative values, depending on the exponent's parity, which complicates its interpretation in real-world applications. Thus, exponential functions are typically defined with a positive base for meaningful real-valued outputs.
In exponential growth functions the base of the exponent must be greater than 1. How would the function change if the base of the exponent were 1 How would the function change if the base of the expon?
"The base of the exponent" doesn't make sense; base and exponent are two different parts of an exponential function. To be an exponential function, the variable must be in the exponent. Assuming the base is positive:* If the base is greater than 1, the function increases. * If the base is 1, you have a constant function. * If the base is less than 1, the function decreases.
Why does b have to be greater than 0 in an exponential function?
In an exponential function of the form ( f(x) = b^x ), where ( b ) is the base, ( b ) must be greater than 0 to ensure that the function is defined for all real numbers ( x ). If ( b ) were less than or equal to 0, the function would either be undefined (as in the case of negative bases for non-integer exponents) or not exhibit the characteristic growth behavior of exponential functions. Additionally, a positive base guarantees that the function remains continuous and either increases (for ( b > 1 )) or decreases (for ( 0 < b < 1 )), maintaining its fundamental properties.
What determines whether the on a graph of an exponential functions increases or decreases?
An exponential function such as y=b^x increases as x goes to infinity for all values in the domain. That is, the function increases from left to right anywhere you look on the graph, as long as the base b is greater than 1. This is called a "Growth" function. However, the graph is decreasing as x goes to infinity if (a) the opposite value of the input is programmed into the function, as in y=b^-x, or if (b) the base is less than 1, as in y=(1/2)^x.
On a graph what shape doesn't exponential function make?
An exponential function does not create a linear shape on a graph. Instead, it produces a curve that either rises or falls rapidly, depending on whether the base of the exponent is greater than or less than one. The graph is characterized by its continuous and smooth nature, exhibiting either exponential growth or decay. Additionally, it does not form any circular or parabolic shapes, which are seen in other types of functions.
Why are the y-values of an exponential growth function either always greater than or less than the asymptote of the function?
The exponential function is always increasing or decreasing, so its derivative has a constant sign. However the function is solution of an equation of the kind y' = ay for some constant a. Therefore the function itself never changes sign and is MORE?
Can a be less than 0 in an exponential equation?
Yes it can.
Logistic and exponential growth?
A logistic growth will at first approximate an exponential growth - until it approximates the "saturation" value, when it begins to increase less quickly.
How do you tell if its exponential growth or decay?
Exponential growth has a growth/decay factor (or percentage decimal) greater than 1. Decay has a decay factor less than 1.
Advantages and disadvantages of logarithm?
Maps and navigational tools needed a huge number of calculations, including multiplications (or divisions) of numbers with several digits. In pre-computer days it was not easy to get people who could do this with a high degree of accuracy. Logarithms changed multiplication into addition and division into subtraction. These were operations that less skilled clerks were able to perform. This advantage has largely disappeared with the easy availability of calculators. The exponential function is one of the more important functions in advanced mathematics, physics and economics. The logarithm function is the inverse of the exponential function and so has very many applications.
Could there be a quadratic function that has an undefined axis of symmetry?
Yes and this will happen when the discriminant of a quadratic equation is less than zero meaning it has no real roots.
What is happen to the tigers?
they are getting less and less.
