0
votes
1answer
20 views

Graph exponential function

I am having problems understanding why $xe^x + 10e^x$ has two $(x,y)$ intercepts. I understand why there is one $(0,10)$, but am unclear on how to return $(-10,0)$. Any help would be much ...
0
votes
1answer
27 views

Need help creating a special function

I'm creating a special function in a game and needed some help with the maths end of it. Essentially, I need a programmable, non-linear function so that $f(100) = 0$, and $f(0) = 100$ (or some other ...
6
votes
10answers
1k views

how to see the logarithm as the inverse function of the exponential?

I saw here in math.stackexchange some proofs of how the log and exp functions are related to each other, but I want to get an intuition for that. In layman terms, how would you explain the connection ...
2
votes
0answers
37 views

Fractional derivative of exponential function

With the $n$th order derivative ($n$ as a positive integer) of $e^{ax}$ given by $$D^{n}e^{ax}=a^ne^{ax},$$ is the generalized (or fractional) derivative the same? Does it apply for any arbitrary ...
1
vote
1answer
50 views

Rewriting in $y=A_0\cdot e^{at}$

How do you rewrite $y = −8(1.589)^{t − 3}$ in $y=A_0e^{at}$ form for appropriate constants $A_0$ and $a?$ For other problems I took the $\ln$ of the number inside the parenthesis. So for example I ...
1
vote
1answer
43 views

Biasing sigmoid curve

I wish to use the sigmoid function $1-{1\over1+e^{-x+c}}$ to obtain a value from 0 to 1 (to be used for a probability value), where $c$ is a constant. The higher this constant, the lower the ...
4
votes
3answers
68 views

Game With 21 Squares, How Many Possible Answers? Function Building

We played this game in our math class, okay, I'll explain how it's played. There are 21 squares in a straight line across, the first person shades in 2 adjacent squares. The next player shades in 2 ...
0
votes
2answers
24 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
1
vote
3answers
135 views

Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.

Prove that the function $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$. My work so far: $f(0)=0$ Thus, $x=0$ is a root. For the ...
-3
votes
1answer
31 views

Can I have an exponential function such that if x = infinity, y = 100?

I tried the most basic y = 100*constant^(1/x) assuming that 1/x = 0 when x is infinity, but it doesn't seem to work. This gives me a function that starts with a higher value of y and goes down till ...
0
votes
1answer
35 views

Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
0
votes
1answer
45 views

Inverse Function of Logarithm

The answer is A but I don't understand why! $ -2 \log_e (x^2) $ can be re-written as $ -4 \log_e(x) $ right? but why do these two graphs look different? the graph $-2 \log_e (x^2) $ is one to ...
0
votes
1answer
17 views

Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
0
votes
2answers
198 views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
0
votes
1answer
64 views

Steps to Graph Exponential Equations & Absolute Value

how to sketch: $-e^{|-x-1|} + 2$ Can someone clarify: $|f(x)|:$ we draw $f(x)$ and then reflect the ($-y$ parts) in the $x$-axis $f|(x)|:$ we draw $f(x)$ and then reflect the ($-x$ parts) in the ...
1
vote
1answer
30 views

Show that $g(x)=x\ln{x}$ and $g(x)=e^x$ are bounded below.

Show that $g(x)$ is bounded below, for $0\leq x$: a) $g(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \\ x\ln{x} & \mbox{if } x>0 \end{array} \right.$ b) $g(x)=e^x$ For (a), ...
0
votes
4answers
71 views

$f:\mathbb R \to (0,\infty)$ defined by $f(x)=e^x$. Describe its inverse.

How do I go about describing it? Well first is the inverse $e^{-x}$ or $\ln(x)$? Additionally, since I have no clue how to solve these problems as I am probably overthinking them... $f:\mathbb R\to ...
0
votes
1answer
37 views

Convergence rate of exponential function

If I have two exponential function, say $f_1(t)=4e^{-3t}+6e^{-7t}$ and $f_2(t)=\frac{2e^{-3t}+5e^{-7t}}{e^{-3t}+9e^{-7t}} - 2$ who are all converge to $0$. Then, the convergence rate of $f_1(t)$ can ...
1
vote
0answers
38 views

Sigmoid Function Question

Ive been trying for well over a week to try to understand how to use a simple sigmoid or logistic function works. Specifically I'm trying to understand how to build proper polynomia parameters for ...
1
vote
2answers
44 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
2
votes
2answers
53 views

Is it possible to rewrite floor functions applied to a fraction using only the addition, multiplication, and exponentiation operators?

Let's restate this question in using mathematical notation. Let $n,k \in \mathbb{N}$. Let $f(n)=\left\lfloor{\frac{n}{k}}\right\rfloor$. Is it possible to rewrite this using the addition, ...
1
vote
3answers
105 views

Is x^x an exponential function?

I know that functions of the form $c^x$ are called exponential when $c$ is a constant. How about the function $x^x$? It seems somewhere in between exponential and double exponential to me. Is there a ...
6
votes
2answers
137 views

Gamma Type Integral

I was hoping someone could help me with a question I came across recently: essentially it's a gamma type integral that your asked to evaluate/reduce: ...
1
vote
1answer
109 views

Proving that linear combination of exponentials is positive

I found the following question in a book without any proof. Question : Prove that $$f(t)=3-5e^{-2t}+6e^{-3t}+2e^{-5t}-3e^{-(3-\sqrt5)t}-3e^{-(3+\sqrt5)t}\gt0$$ for any $t\gt0$. The book says that ...
0
votes
2answers
82 views

What is the simplest $\Bbb{R}\to\Bbb{R}$ function with two peaks and a valley?

What is the simplest $\Bbb{R}\to\Bbb{R}$ function with two peaks and a valley? I have a set of points in $\Bbb{R^2}$ and I would like to fit a curve to the points, the points approximately lie on a ...
1
vote
1answer
108 views

How to define a surface $z = f(x,y)$ with flat region at centre and sigmoidally tapering towards the edges?

How do we define a continuos function $f(x,y)$ within the bounded domain $x \in [a,b]$ and $y \in [c,d]$ so that $z=f(x,y)$ has a flat surface at the centre (flat means $f(x,y)= C$, $C$ being ...
3
votes
2answers
712 views

When is the sum of two exponentials functions equal to another exponential function?

Fix real numbers $a_1$, $a_2$, $a_3$ and $b_1$, $b_2$, $b_3$ and $c_1$, $c_2$, $c_3$. Consider the equation $$ a_1\exp(b_1 (x-c_1)) + a_2\exp(b_2 (x-c_2)) = a_3\exp(b_3 (x-c_3)) $$ in $x$. My ...
2
votes
1answer
72 views

Is there a name for the function $(1 - e^{ct})/(1 - e^{c})$?

$$f(t) = \frac{1 - e^{ct}}{1 - e^{c}}$$ This is a function which is somehow a streched exponential which is zero at $t = 0$, and one at $t = 1$, where $c$ determines the curvature (with $c = 0$, it ...
5
votes
2answers
603 views

Help finding inverse of $f(x)=\frac{e^x-e^{-x}}{2}$

I'm trying to find the inverse of $f(x)=\frac{e^x-e^{-x}}{2}$. My textbook says $f^{-1}(x)=\ln(x+\sqrt{x^2+1})$, but I haven't been able to get that answer. Switching $x$ and $y$, I tried solving for ...
1
vote
1answer
867 views

How would I create a exponential ramp function from 0,0 to 1,1 with a single value to explain curvature?

I need an exponential function that will take linear input from 0,0 to 1,1 and give me back an exponential shaped curve such that changes in X near the 0 point result in small increases in Y, but each ...
3
votes
3answers
6k views

The relation between an exponential function and a logarithmic function

I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're ...
0
votes
1answer
2k views

Exponential and power functions through two points

I have a problem where I'm asked to determine the constants of exponential and power functions that go through both points (5, 50) and (10, 1600). I have tried to solve them below, but would ...
0
votes
1answer
103 views

How do I solve this exponential equation?

$$x = 2^{x-3}$$ Does there exist an analytical solution to this equation? If so, how do I find it? What if it is changed to an equality? $$x>2^{x-3}$$
2
votes
1answer
156 views

Reflecting an exponential function over a y = 3 line.

How would you write the equation of $f(x) = 4^x$ that reflects over the line $y = 3$? I've put in $f(x) = 3 + 4^{-x}$ which I thought was the right answer, but it isn't. Thanks in advance!
2
votes
2answers
217 views

Exponential growth function

Under ideal conditions a certain bacteria population is know to double every three hours. Suppose that there are initially 100 bacteria. How would you go about formulating a function for this?