Is This Function An Exponential

If f(x) is an exponential function where f(4)=2 and f(7.5)=72, then how do you find the value of f(9.5), to the nearest hundredth? What is the value of f(9.5)?

There are three ways I can think of to do this, but here is the easiest:Input x linearOutput f(x) exponentialwhen x increases by 3.5(to go from 4 to 7.5) output increases by a FACTOR of 36 to go from 2 to 72.An increase in x of 1 unit gives us a factor of 36^(1/3.5) or approx 2.783924So when x increases by 5.5(to go from 2 to 9.5), f(x) is MULTIPLIED by factor 2.783924^5.5 = 279.00742 x 279.0074 = 558.01 to the nearest hundredth as you specified:).If you actually meant to the nearest hundred we can take the answer to be 600:)Of course you can say thatf(x) = Ae^(kx) and input your x and y = f(x) values to get two equations in A and k. Then eliminate k to find A. And substitute back to find k. And then input 9.5. But it takes much longer. And yes, you will get the same answer:)You can also use the compound interest formula, since CI is exponential growth, and I’ll leave that for an exercise you might like to try for yourself.

What are exponential functions?

Exponential functions are functions of the form f(x) = b^x where b is a constant.The real mathematical importance of exponential functions is in their being proportional to their derivatives meaning the bigger x is, the steeper the slope of the function. This means they grow extremely fast: exponentially fast.A common example of exponential growth is a bacterial population. Since bacteria reproduce rapidly by mitosis, for every bacterium in a culture, there will, before long be two bacteria. Thus, if I have 10 bacteria to start, one minute later I will have 20, and in another minute, each of those twenty will have reproduced and I will have 40; one minute later the same will have happened again and before long, the culture will be overrun with bacteria. The more bacteria there are, the more bacteria there are to make even more bacteria the next minute.All exponential functions are proportional to their derivatives, but the function f(x) = e^x (where e = 2.718281828459…) is actually equal to its derivative meaning the y value of the function at a particular x value is equal to the slope of the curve at that same x value. Thus when dealing with exponential functions in calculus, e becomes a very natural number to use as a base. This is why e is such a big deal and why teachers try to introduce the number early on in Algebra 2, even though it doesn’t have much use at that point in one’s mathematical education.When plotting an exponential curve by hand, one ought to simply pick whole number x values and multiply the base by itself an x number of times to calculate the corresponding y value. This will give a rough plot of the graph of the curve, although a calculator is requires to plot the points not on whole number x values. The curve of simple exponential functions is always increasing and concave up, or vice versa for b^-x. For a positive exponential function like f(x) = e^x, as x approaches ∞, f(x) approaches ∞, and as x approaches -∞, f(x) approaches 0. The curve has no absolute maximum or minimum value and has a horizontal asymptote at y = 0.

Is an exponential function even or odd?

The exponential function [math]f([/math]x[math])=e^x[/math] is neither nor odd.Because [math]f(-x)=e^{-x} [/math]which is not equal to either [math]f(x) [/math] [math]or -f(x) [/math]Rather, An exponential function is a sum of an even and an odd fuction. The even fuction is known as cosh x and odd function is known as sinh x.Let us explain it. We can write[math]e^x=e^x[/math][math]e^x=\frac{1}{2} (2e^x) [/math][math]e^x=\frac{1}{2} (e^x+e^x) [/math][math]e^x=\frac{1}{2} (e^x+e^{-x} +e^x-e^ {-x} ) [/math][math]e^x=\frac{e^x+e^{-x}} {2}+\frac{e^x-e^{-x} }{2}[/math][math]e^x=cosh x+ sinh x[/math][math];cosh x=\frac{e^x+e^{-x}}{2} , sinhx=\frac{e^x-e^{-x}}{2} [/math]cosh x is even function and sinx is odd function.

Are exponential functions one to one?

That depends on your domain. Assuming your domain is the reals, then yes, and here’s why.A function is one-to-one if and only if and only if[math] f(x) = f(y) \implies x = y [/math].This essentially captures the idea, only one input can produce a particular output, in a rigorous definition. Now for the simplest exponential function[math] f(x) = e^x [/math].We can use the fact that if [math] x < y [/math], then [math] e^x < e^y [/math] (this can be easily proven given any definition of the exponential function).Now suppose we have two number [math] x, y [/math], such that [math] e^x = e^y [/math]. We know [math] x [/math] cannot be less than [math] y [/math], otherwise [math] e^x < e^y [/math], so we can say [math] x \geq y [/math]. Moreover, we know [math] x [/math] cannot be greater than [math] y [/math] for a similar reason, so we have [math] x \leq y [/math]. Putting these together, we have [math] x = y [/math], which completes the proof.Now for a general exponential function [math] f(x) = Ae^{kx} [/math], its enough to use the fact that [math] e^x [/math] is one-to-one (which we just proved), and that linear functions and composition of one-to-one functions are one-to-one (both of these should be obvious).Note that we need this fact to justify the use of logs over the reals as functions (logs are defined as the inverse of exponential functions, which only works if exponential functions are one-to-one).Also note that over the complex numbers, [math] e^x [/math] is not one-to-one (consider [math] e^{i\pi} = e^{3i\pi} = -1 [/math]). This causes problems when trying to take logs of complex numbers, similar to when you invert a trigonometric function.