Question

Tell whether the function \(f(x)=\frac{1}{3} e^{4 x}\) represents exponential growth or exponential decay. Explain.

Short answer

The function \(f(x)=\frac{1}{3} e^{4 x}\) represents exponential growth because the base of the exponent is greater than 0.

Step-by-step solution

Identify the function and its components

The given function is of the form \(y=a e^{bx}\), a standard form for an exponential function, with \(a=\frac{1}{3}\) and \(b=4\).

Determine Nature of Exponential Function

To determine if this function shows exponential growth or decay, examine the value of \(b\) in \(y=a e^{bx}\). If \(b>0\), the function represents exponential growth. If \(b<0\) then it represents exponential decay. In this case, \(b=4\) which is greater than zero.

Conclusion

Since the value of \(b\), the exponent base, in the function is greater than 0, the function \(f(x)=\frac{1}{3} e^{4 x}\) represents exponential growth.

Key concepts

Exponential Growth

When we talk about exponential growth in mathematics, we refer to a situation where a quantity increases at a rate proportional to its current value. This type of growth can be incredibly rapid, and it often occurs in real-life scenarios such as population growth, compound interest, and certain types of infectious disease spread.

For a function like \(f(x) = \frac{1}{3}e^{4x}\), where the exponent of \(e\) contains a positive constant multiplied by \(x\), you can expect that as \(x\) gets larger, the value of \(f(x)\) will increase at an exponential rate. This is because the positive exponent implies that the input variable \(x\) is being used to exponentiate a number greater than 1—in this case, Euler's number \(e\), which is approximately 2.718. This exponentiation results in the function growing bigger as \(x\) increases.

To understand exponential growth easily, imagine each step you take is twice as large as the previous step. With each step, the distance you cover escalates rapidly, getting exponentially larger. Such is the nature of exponential growth in functions like the one given in the exercise.

Exponential Decay

Exponential decay is the opposite of exponential growth. It describes a process where a quantity decreases at a rate proportional to its current value. This type of behavior is typical in phenomena such as radioactive decay and cooling of hot objects.

In an equation form, a decaying exponential function has a negative exponent, represented typically as \(f(x) = ae^{-bx}\), where \(a\) is a positive constant, \(b\) is a positive rate of decay, and \(e\) is Euler's number. The negative sign in the exponent signifies that the value of \(f(x)\) decreases as \(x\) increases. Therefore, for any positive value of \(b\), the function's value diminishes over time. It's like having a pie that gets smaller each time you take away a portion that's proportional to the size of the remaining pie—it never quite disappears but gets ever closer to being gone.

Exponential Function Components

An exponential function can be recognized in its general form of \(y = ae^{bx}\), where 'e' is the base of the natural logarithm, and 'a' and 'b' are constants that determine the function's behavior. Let’s break down the components:

• \(a\) — This is the initial value or the y-intercept when \(x = 0\). It's the starting point for the exponential curve on a graph. If \(a\) is positive, the graph will be positioned above the x-axis; if it's negative, the graph will be below.
• \(b\) — Known as the growth or decay rate, if this constant is positive, the function will exhibit exponential growth. If it's negative, the function will exhibit exponential decay.
• \(e\) — Euler's number (approximately 2.718), which is a mathematical constant used as the base for natural logarithms.

The 'x' is the variable that, when increases, the function's value will change depending on the signs and values of 'a' and 'b'. In exploring these components, students gain a strong foundation for understanding how exponential functions behave. Identifying and manipulating these components will allow students to solve a wide range of problems involving exponential relationships.