Exponential Function
A video goes viral: every hour, the number of people who have watched it is multiplied by a fixed factor, not increased by a constant amount. A cup of coffee cools down, and the difference between its temperature and the room's temperature shrinks by the same proportion every minute. In both cases the variable $t$ is not in the base but in the exponent, and that completely changes how things grow or decay. This is the function you will study here.
The Exponential Function
Every function $f(x)=a^x$ passes through $(0,1)$, because $a^0=1$, and through $(1,a)$, because $a^1=a$. The point $(0,1)$ is therefore the $y$-intercept of all exponentials. Move the base and observe what stays fixed.
Among all the bases there is a special one: the number $e\approx2.71828\ldots$, which is irrational and shows up in continuous growth processes (continuously compounded interest, radioactive decay, cooling). The function $e^x$ is the reference exponential in science and engineering. Its base is left written as $e$ unless a numerical value is requested.
Given the exponential function $f(x) = 3^x$, calculate the numerical values of $f(2)$, $f(-1)$, $f(0)$ and $f(\frac{1}{2})$. Express the results in exact form, using…
Given the exponential function $f(x) = 3^x$, calculate the numerical values of $f(2)$, $f(-1)$, $f(0)$ and $f(\frac{1}{2})$. Express the results in exact form, using roots or fractions if necessary.
Find the algebraic expression for an exponential function of the form $f(x) = k \times a^x$, knowing that its graph passes through the points $(0, 3)$ and $(2, 12)$.
Find the algebraic expression for an exponential function of the form $f(x) = k \times a^x$, knowing that its graph passes through the points $(0, 3)$ and $(2, 12)$.
Determine the domain of definition of the following function: $f(x) = \frac{x+2}{e^x - e^2}$
Determine the domain of definition of the following function:
$f(x) = \frac{x+2}{e^x - e^2}$
Increasing and Decreasing Behavior
Why? When going from $x$ to $x+1$, the value is always multiplied by $a$: $\dfrac{f(x+1)}{f(x)}=a$. If that factor is greater than $1$, each step increases the value; if it is less than $1$, each step reduces it.
The graph of $-a^x$
If you put a minus sign in front, $f(x)=-a^x$, every value changes sign: the graph is the reflection across the $x$-axis of the graph of $a^x$. Now it passes through $(0,-1)$, all its values are negative (range $(-\infty,0)$), and it approaches $y=0$ from below. If $a>1$, $-a^x$ is decreasing and goes down toward $-\infty$ as $x\to+\infty$; if $0 Classify the following exponential functions as increasing or decreasing, justifying your answer based on the value of their base: $f(x) = 4^x$ $g(x) = \left(\frac{1}{3}\right)^x$ $h(x) = 0.8^x$ $i(x) = e^x$ Match each of the graphs labelled A, B, C and D in the following figure with its corresponding exponential function. The functions are: $f(x) = 2^x$, $g(x) = 0.5^x$, $h(x) = 3^x$ and $i(x) = -2^x$. The attached graph shows the functions $f(x) = e^x$ and $g(x) = e^{-x}$, together with the horizontal line $y = e$. Determine analytically the coordinates of the point of intersection between the functions $f(x)$ and $g(x)$. Then, calculate the exact area of the triangle whose vertices are this point of intersection and the two points where the curves intersect the line $y = e$.Classify the following exponential functions as increasing or decreasing, justifying your answer based on the value of their base: $f(x) = 4^x$…
Match each of the graphs labelled A, B, C and D in the following figure with its corresponding exponential function. The functions are: $f(x) = 2^x$, $g(x) = 0.5^x$,…
The attached graph shows the functions $f(x) = e^x$ and $g(x) = e^{-x}$, together with the horizontal line $y = e$. Determine analytically the coordinates of the point…
Domain, Range, and Horizontal Asymptote
When you translate the graph, the asymptote is translated along with it. In $g(x)=a^{x-p}+q$:
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The exponent $x-p$ shifts the curve $p$ units horizontally.
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The final $+q$ shifts it $q$ units vertically.
If $a>1$, as $x\to-\infty$ the term $a^{x-p}$ tends to $0$, so $\lim_{x\to-\infty}\left(a^{x-p}+q\right)=q$. The asymptote is $y=q$. If $0 Determine the domain, range and the equation of the horizontal asymptote of the function $f(x) = 2^x - 5$. Calculate analytically the coordinates of the x-intercept and y-intercept of the function $f(x) = 3^{x+1} - 9$. Starting from the function $f(x) = 2^x$, explain in detail what geometric transformations must be applied to its graph to obtain the graph of the function $g(x) = 2^{x-3} + 1$. Determine the domain, range and the equation of the horizontal asymptote of $g(x)$.Determine the domain, range and the equation of the horizontal asymptote of the function $f(x) = 2^x - 5$.
Calculate analytically the coordinates of the x-intercept and y-intercept of the function $f(x) = 3^{x+1} - 9$.
Starting from the function $f(x) = 2^x$, explain in detail what geometric transformations must be applied to its graph to obtain the graph of the function…
Vertical Asymptotes, Sign, and Continuity
When the exponent is an expression in $x$, as in $f(x)=e^{g(x)}$, the exponential inherits the problems of $g$. Three ideas are enough.
A vertical asymptote $x=c$ appears where the function "shoots up" as you approach $c$. To find out, you compute the one-sided limits: $x\to c^-$ (from the left) and $x\to c^+$ (from the right). It is enough for one of them to be infinite.
To study the sign of a function with an exponential factor, remember that $e^{g(x)}>0$: the sign is decided by the other factor.
Finally, $a^x$ is continuous on all of $\mathbb{R}$. In a piecewise function with exponential pieces, continuity is decided at the points where the formula changes: the one-sided limits must match the value of the function. For example, if a piece $e^{ax}+b$ is joined at $x=0$ with $3x+2$, you must set $e^{0}+b=1+b$ equal to $3\cdot0+2=2$.
Determine analytically the intervals of constant sign (that is, the intervals where the function is strictly positive and where it is strictly negative) of the function:…
Determine analytically the intervals of constant sign (that is, the intervals where the function is strictly positive and where it is strictly negative) of the function:
$f(x) = (x^2 - 5x + 6)e^{-x}$
Given the function $f(x) = e^{\frac{2x}{x-1}}$, carry out an analytical study to determine: Its domain of definition. Its axis intercepts. The equations of its…
Given the function $f(x) = e^{\frac{2x}{x-1}}$, carry out an analytical study to determine:
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Its domain of definition.
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Its axis intercepts.
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The equations of its horizontal and vertical asymptotes, if it has any.
Find the values of the parameters $a$ and $b$ so that the following piecewise function is continuous on all of $\mathbb{R}$: $$ f(x) = \begin{cases} e^{ax} + b &…
Find the values of the parameters $a$ and $b$ so that the following piecewise function is continuous on all of $\mathbb{R}$:
$$ f(x) = \begin{cases} e^{ax} + b & \text{if } x < 0 \\ 3x + 2 & \text{if } 0 \le x \le 2 \\ 2^{x-1} + a & \text{if } x > 2 \end{cases} $$Exponential Equations
There are four techniques, from the simplest to the most general.
Why do logarithms work here? Because the logarithm is the inverse function of the exponential: $\log_a x$ undoes what $a^x$ does, and vice versa. That is why $\log_a(a^x)=x$ and $a^{\log_a x}=x$; with base $e$, $\ln(e^x)=x$ and $e^{\ln x}=x$. On the graph, the curves $y=a^x$ and $y=\log_a x$ are reflections of each other across the line $y=x$: if $(p,q)$ is on one, $(q,p)$ is on the other.
Systems of exponential equations
When there are two unknowns in exponents and the bases are different, give each power a new letter, $u=2^x$ and $v=3^y$ (with $u>0$, $v>0$). The exponential system becomes a linear system in $u$ and $v$. You solve it with any of the methods you already know (substitution, equating, elimination) and at the end you undo the substitution.
When the numbers are chosen so that exact powers come out (for example $u=8$ and $v=9$), undoing the substitution is immediate: $2^x=8\Rightarrow x=3$ and $3^y=9\Rightarrow y=2$.
Solve the following exponential equation by equating the bases on both sides of the equation: $5^{2x-1} = 125$
Solve the following exponential equation by equating the bases on both sides of the equation:
$5^{2x-1} = 125$
Express both sides of the equation as powers with base $2$ and solve: $8^x = \frac{1}{4}$
Express both sides of the equation as powers with base $2$ and solve:
$8^x = \frac{1}{4}$
Solve the following exponential equation by making an appropriate substitution: $9^x - 10 \times 3^x + 9 = 0$
Solve the following exponential equation by making an appropriate substitution:
$9^x - 10 \times 3^x + 9 = 0$
Solve the following exponential equation using logarithms: $4^{x-1} = 15$ Express the result in exact form using logarithms and also provide an approximation rounded to…
Solve the following exponential equation using logarithms:
$4^{x-1} = 15$
Express the result in exact form using logarithms and also provide an approximation rounded to two decimal places.
Solve the following exponential equation by factorising out a common factor after applying index laws: $2^{x+2} + 2^{x-1} - 2^{x-2} = 34$
Solve the following exponential equation by factorising out a common factor after applying index laws:
$2^{x+2} + 2^{x-1} - 2^{x-2} = 34$
Solve the following system of exponential equations: $$ \begin{cases} 2^x + 3^y = 17 \\ 2^{x+1} - 3^{y-1} = 13 \end{cases} $$
Solve the following system of exponential equations:
$$ \begin{cases} 2^x + 3^y = 17 \\ 2^{x+1} - 3^{y-1} = 13 \end{cases} $$Growth and Decay Models
When the number of compounding periods per year grows without bound, you get continuously compounded interest, $C(t)=C_0e^{rt}$. It has the same form $e^{\pm kt}$ as Newton's cooling or decay measured with the constant $k$.
A population of bacteria doubles every hour. If there are initially $150$ bacteria in the culture, write the function $P(t)$ that models the number of bacteria as a…
A population of bacteria doubles every hour. If there are initially $150$ bacteria in the culture, write the function $P(t)$ that models the number of bacteria as a function of time $t$ (expressed in hours). How many bacteria will be in the culture after $4$ hours?
An amount of $100\,000$ pesos is invested in a bank deposit offering an annual compound interest rate of $3\%$. Write the function that determines the accumulated…
An amount of $100\,000$ pesos is invested in a bank deposit offering an annual compound interest rate of $3\%$.
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Write the function that determines the accumulated capital $C(t)$ after $t$ years.
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Calculate the total capital that will have accumulated after $10$ years.
A radioactive isotope has a half-life (the time it takes for its initial mass to be reduced by half) of $8$ days. Starting from an initial sample of $200$ grams: Write…
A radioactive isotope has a half-life (the time it takes for its initial mass to be reduced by half) of $8$ days. Starting from an initial sample of $200$ grams:
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Write the function $M(t)$ that gives the remaining mass of the isotope as a function of time $t$ in days.
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What mass of this isotope will remain after $24$ days?
The market value of an industrial machine decreases exponentially over time. It was bought for $800\,000$ pesos and it is known that each year it loses $15\%$ of its…
The market value of an industrial machine decreases exponentially over time. It was bought for $800\,000$ pesos and it is known that each year it loses $15\%$ of its value relative to the immediately preceding year.
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Write the function $V(t)$ that gives the value of the machine as a function of the years elapsed.
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What will its value be after $5$ years?
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Exactly how much time must elapse for its value to be halved?
The spread of a virus in an isolated population of $10000$ inhabitants follows a logistic growth model given by the function: $P(t) = \frac{10000}{1 + 99 e^{-0.5t}}$…
The spread of a virus in an isolated population of $10000$ inhabitants follows a logistic growth model given by the function:
$P(t) = \frac{10000}{1 + 99 e^{-0.5t}}$
where $P(t)$ is the number of infected people and $t$ is the elapsed time in days from the start of the outbreak.
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Calculate the initial number of infected people at time $t = 0$.
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Determine how many people will be infected after $10$ days.
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Calculate the exact time required for half of the population to be infected.
According to Newton's law of cooling, the temperature of an object at time $t$ (in minutes) is given by the expression $T(t) = T_a + (T_0 - T_a)e^{-kt}$, where $T_a$ is…
According to Newton's law of cooling, the temperature of an object at time $t$ (in minutes) is given by the expression $T(t) = T_a + (T_0 - T_a)e^{-kt}$, where $T_a$ is the ambient temperature and $T_0$ is the initial temperature of the object. A cup of coffee at $90^\circ\text{C}$ is left in a room kept at a constant temperature of $20^\circ\text{C}$. If after $10$ minutes the temperature of the coffee has dropped to $60^\circ\text{C}$:
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Calculate the exact value of the cooling constant $k$.
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Determine the total time elapsed from the beginning for the coffee to reach a temperature of $30^\circ\text{C}$.
Back to the beginning: if the video has $N_0$ views and is multiplied by $k$ every hour, the views follow $N(t) = N_0\, k^t$. And if a quantity is cut in half in every interval, it follows the decreasing model $V(t) = V_0\left(\frac12\right)^t$.