Exponential Function
Yaritza posts a video on social media and shares it with 2 friends. Each of them shares it with 2 more people, and so on: in each round the number of people who receive the video is multiplied by 2, not increased by a fixed amount. With a cell phone and just a few rounds, it reaches thousands of people. In this situation the variable (the rounds) is in the exponent, and that completely changes how things grow. This is the exponential function.
The exponential function
Every function $f(x)=a^x$ passes through $(0,1)$, because $a^0=1$, and through $(1,a)$, because $a^1=a$. Move the slider and check that the graph always passes through $(0,1)$.
Among all the bases there is a special one: the number $e\approx2.71828\ldots$, an irrational number that appears in continuous growth processes (continuous interest, radioactive decay, cooling). We leave it written as $e$ unless the problem asks for an approximate value.
To work with unknown exponents we will use the natural logarithm, $\ln$, which satisfies $\ln e=1$ and $\ln(a^x)=x\ln a$. The logarithmic function section studies it in depth; here we use it only as a tool.
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:
$f(x) = \frac{x+2}{e^x - e^2}$
Growth and decay
The reason: going from $x$ to $x+1$, the value is always multiplied by $a$, that is, $\dfrac{f(x+1)}{f(x)}=a$. If that factor is greater than $1$, each step increases the value; if it is less than $1$, it reduces it.
Reflection: the function $-a^x$
If you put a minus sign in front, $f(x)=-a^x$, every value changes sign: the graph of $a^x$ is reflected across the $x$-axis. It passes through $(0,-1)$, is always negative, its range is $(-\infty,0)$, and the asymptote is still $y=0$. If $a>1$, $-a^x$ is decreasing: it drops faster and faster.
This is how you tell four typical graphs apart: $2^x$ (increasing, above), $3^x$ (increasing, steeper than $2^x$), $0.5^x$ (decreasing, above), and $-2^x$ (negative, below). First look at the sign (above or below the $x$-axis) and then at the base.
Classify the following exponential functions as increasing or decreasing, justifying your answer based on the value of their base: $f(x) = 4^x$…
Classify the following exponential functions as increasing or decreasing, justifying your answer based on the value of their base:
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$f(x) = 4^x$
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$g(x) = \left(\frac{1}{3}\right)^x$
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$h(x) = 0.8^x$
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$i(x) = e^x$
Match each of the graphs labelled A, B, C and D in the following figure with its corresponding exponential function. The functions are: $f(x) = 2^x$, $g(x) = 0.5^x$,…
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…
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 $y = e$.
Asymptote and translations
When the graph is translated, the asymptote is translated with it. In $g(x)=a^{x-p}+q$:
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The exponent $x-p$ shifts the curve $p$ units horizontally (to the right if $p>0$).
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The final $+q$ shifts it $q$ units vertically.
If $a>1$, when $x\to-\infty$ the term $a^{x-p}\to0$, so $\lim_{x\to-\infty}\left(a^{x-p}+q\right)=q$. The horizontal asymptote is $y=q$, the domain is still $\mathbb{R}$, and the range is $(q,+\infty)$. (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…
Exponential equations
An exponential equation has the unknown in the exponent. There are four basic techniques.
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:
$9^x - 10 \times 3^x + 9 = 0$
More equations and systems
An exponential system is solved with a change of variable for each power: $u=2^x$, $v=3^y$. This leaves a linear system.
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
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 $300\,000$ 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 $300\,000$ 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?
Problems with base e
When growth or decay is continuous, the model uses $e^{kt}$. To find the time or the constant, solve using $\ln$.
The market value of an industrial machine decreases exponentially over time. It was bought for $2\,400\,000$ 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 $2\,400\,000$ 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 $30^\circ\text{C}$.
Analytical study of functions with exponentials
To study a function that contains $e^{(\ldots)}$ you use three ideas.
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Since $e^u>0$ always, it never equals zero and does not change the sign of a product: the sign is decided by the other factor.
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The domain comes from the exponent (a fraction cannot have a denominator of $0$).
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The asymptotes come from the limits of the exponent: if $u\to c$, then $e^u\to e^c$.
A piecewise function is continuous if at each joining point the one-sided limits and the value of the function coincide.
With two joining points you get two equations, and they are solved together (with two unknowns). In the continuity exercise, use $x=0$ and $x=2$.
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} $$Back to the beginning: if Yaritza shares the video with 2 friends and each round is multiplied by $2$, in round $t$ it is received by $N(t)=2^t$ people: in round 10 that is already $2^{10}=1{,}024$.