Problem 241 - Calculus Volume 1

Question

Sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote. \(\quad f(x)=3^{x+1}\)

Step-by-Step Solution

Answer

Domain: all real numbers; Range: positive numbers; Horizontal Asymptote: y=0.

Step 1: Understand the function

The given function is an exponential function of the form \( f(x) = a^{x+b} \), where \( a=3 \) and \( b=1 \). The function is increasing since the base \( a=3 \) is greater than 1.

Step 2: Identify the horizontal asymptote

For exponential functions \( f(x) = a^{x+b} \), the horizontal asymptote is the line \( y=0 \). This is because, as \( x \to -\infty \), \( f(x) \to 0 \).

Step 3: Determine the Domain

The domain of an exponential function is all real numbers. Therefore, for \( f(x) = 3^{x+1} \), the domain is \( (-\infty, \infty) \).

Step 4: Determine the Range

Since the exponential function is of the form \( f(x) = a^{x+b} \), it will only take positive values. Thus, the range of \( f(x) = 3^{x+1} \) is \( (0, \infty) \).

Step 5: Sketch the graph in the coordinate plane

Start by plotting the y-intercept by setting \( x=0 \). \( f(0) = 3^{0+1} = 3 \). This will give the point \( (0, 3) \). For \( x=1 \), \( f(1) = 3^{1+1} = 9 \), resulting in point \( (1, 9) \). As \( x \to -\infty \), the approach to the horizontal asymptote \( y=0 \) is visible. This lets you draw the curve showing rapid increase to the right and decreasing towards zero to the left without touching the x-axis.

Key Concepts

Graph Sketching

Graph sketching is a crucial skill when working with exponential functions, as it provides a visual representation of the function's behavior. For the function \( f(x) = 3^{x+1} \), it is of the form \( a^{x+b} \), where \( a = 3 \) and \( b = 1 \). This indicates the following characteristics:

• The graph is an upward sloping curve because the base of the exponent, 3, is greater than 1, indicating an increasing function.
• To plot the graph, identify key points such as the y-intercept. Set \( x = 0 \), resulting in \( f(0) = 3^{0+1} = 3 \), giving the point \( (0, 3) \).
• Another key point is when \( x = 1 \), yielding \( f(1) = 3^{1+1} = 9 \), corresponding to \( (1, 9) \).

These points guide the shape of the curve, which rapidly rises as \( x \) increases and moves closer to the x-axis as \( x \) decreases. Remember, the graph never touches the x-axis.

Function Domain

The function domain refers to all the possible input values \( x \) that can be used in a function. For exponential functions like \( f(x) = 3^{x+1} \), the domain is vast.
Since the exponent part \( x+1 \) can be any real number, \( x \) itself can also be any real number.
Hence, this function has a domain of all real numbers, represented as \( (-\infty, \infty) \).This means you are free to choose any real number for \( x \), making exponential functions excellent choices for modeling continuous growth processes without the restrictions seen in other types of functions.

Function Range

The range of a function is the set of all possible output values. For the function \( f(x) = 3^{x+1} \), this is impacted by the nature of exponential expressions.Exponential functions always yield positive results. For this reason, no matter what real number \( x \) you input, \( f(x) \) will take on positive values.
The smallest value approaches zero but never actually reaches it. Correspondingly, the range of this function is expressed as \( (0, \infty) \), indicating that outputs will be all positive numbers, not allowing zero.This characteristic makes exponential functions suitable for modeling phenomena such as population growth or radioactive decay, where quantities never diminish to or below zero.

Horizontal Asymptote

A horizontal asymptote is a line that the graph of a function approaches but never quite reaches as \( x \) heads towards infinity or negative infinity.For the exponential function \( f(x) = 3^{x+1} \), the horizontal asymptote is \( y = 0 \). This happens because, as \( x \to -\infty \), \( 3^{x+1} \to 0 \).
The function value gets smaller and smaller, approaching zero without ever touching the x-axis.Understanding horizontal asymptotes is crucial when sketching an exponential graph, as it helps predict end-behavior.
This aspect is particularly helpful in fields such as economics and biology, where predicting long-term trends and plateaus is essential.