Question

In Exercises 31-34, use a table of values or a graphing calculator to graph the function. Then identify the domain and range. $$ y=3 e^x-5 $$

Short answer

The graph of \(y = 3e^x - 5\) passes through the y-axis at -2, increases very rapidly for positive x and asymptotically approaches the line y=-5 for negative x. Its domain is all real numbers, or \(-\infty < x < +\infty\), and its range is all numbers greater than -5, or \(-5 < y < +\infty\).

Step-by-step solution

Understand the Function

The function given is in the form of an exponential function. During graphing, keep in mind that for any real number, the value of \(e^x\) is always positive and it grows very fast as \(x\) increases. Multiplied by 3 and subtracting 5 simply scales and translates the basic shape of the \(e^x\) graph.

Plot the Key Points

Plot a few key points for the function \(y = 3e^x - 5. The y-intercept happens when \(x = 0\), which results in \(y = 3e^0 - 5 = 3 - 5 = -2\). Also plot a point for \(x = 1\) which gives \(y = 3e^1 - 5 = 3e - 5\), and for \(x = -1\) yielding \(y = 3e^{-1} - 5\). This gives a general idea of the graph behavior.

Draw the Graph

Using the key points, draw a smooth curve indicating the graph of the function. The graph is above the x-axis for large positive \(x\), it passes through the y-axis at -2, and approaches y=-5 (but never reaches it) for large negative \(x\). It never crosses the horizontal line \(y = -5\), this is an asymptote.

Determine the Domain and Range

The domain of a function refers to all possible input values (in this case, x-values) and the range refers to all resulting output values (y-values). For this graph, the function is defined for all real numbers, so the domain is \( (-\infty, +\infty)\). It can yield all numbers greater than -5, so the range is \((-5, +\infty)\)

Key concepts

Graphing Calculators

Graphing calculators are powerful tools that simplify the process of graphing complex functions like exponential functions. By using a graphing calculator, you can quickly visualize how a function behaves over a range of values without manually plotting each point. This is especially useful for functions like \[ f(x) = 3e^x - 5 \], which includes both scaling and translation.

When using a graphing calculator to graph such a function, there are a few steps you should follow:

• Input the function: Enter the equation \( y = 3e^x - 5 \) into the calculator's function entry mode.
• Setup the viewing window: Adjust the x and y limits of the graph to ensure that you capture the interesting parts of the function, where significant changes occur.
• Analyze the graph: Observe the behavior of the curve, particularly the position of the asymptote at \( y = -5 \) and how the graph rapidly increases.

Using a graphing calculator makes it easier to comprehend the overall trends and properties of the function, such as its asymptotic behavior and rapid growth as \( x \) increases.

Domain and Range

Understanding the domain and range of a function is crucial when you are studying functions like \( y = 3e^x - 5 \). The domain refers to all the possible input values (x-values), while the range includes all the possible output values (y-values) the function can produce.

For exponential functions, like the one in question, the domain is always all real numbers. This means that you can plug in any real number for \( x \) and get a valid result. Therefore, the domain of our function is \( (-\infty, +\infty) \).

The range, on the other hand, is determined based on the possible output values. Since \( 3e^x \) is always positive and greater than zero, subtracting 5 means the result will always be greater than -5. Thus, the range of the function \( y = 3e^x - 5 \) is \( (-5, +\infty) \), indicating that the output of the function is never less than -5 but can go infinitely high as \( x \) increases.

Asymptotes

An asymptote is a line that a graph approaches but never actually reaches. They are an important concept in understanding the behavior of functions, especially exponential functions.

For the function \( y = 3e^x - 5 \), there is a horizontal asymptote at \( y = -5 \). As \( x \) moves towards negative infinity, the graph gets closer and closer to the line \( y = -5 \) but never actually touches it. This is because \( 3e^x \) approaches zero but is never exactly zero, hence subtracting 5 will keep \( y \) slightly above \(-5\).

Horizontal asymptotes like this one help in visualizing the long-term behavior of functions. In this context, knowing where the asymptote lies allows you to predict that no matter how \( x \) changes, the value of \( y \) will not drop below \( -5 \), no matter how large \( x \) becomes negative. As \( x \) increases, \( y \) grows increasingly faster, distancing itself further from the asymptote.