Analyzing a function means explaining how you determined all of the function’s features, characteristics, and trends.

It is communication for other people to understand.

\(\blacktriangleright \) Domain
First, any function analysis would discuss the domain. Perhaps the domain is already stated with the function definition. Otherwise, the function might be described with a formula. Then, the natural or implied domain can be deduced. If the function is defined via a graph, then the implied domain can be deduced visually. A function might be defined via a function equation relating it to another function. In this case, the equation would help us figure out the induced domain.

If a formula is provided, then we might categorize the function and obtain informaiton about the domain. Otherwise, we could look for numbers that would cause problems with the formula.

If a graph is the only information presented, then there is an unavoidable bubble of inaccuracy to work within. The nature of drawing, the tools used and the person drawing, make inaccuracy simply a part of graphing or decerning information from a graph.

Even with a formula, piecing the graph together would give helpful clues.

We want exact values when we can get them. Otherwise, we settle for approximations. But we want algebra first. Then we settle.

\(\blacktriangleright \) Zeros
Identify all zeros. This will involve solving equations that you create from your knowledge of the elementary functions. If a graph is the only information available, then look for intercepts.

\(\blacktriangleright \) Continuity
The domain, discontinuities, and singularities should begin establishing intervals of continuity. The endpoints or isolation numbers may require additional scrutiny.

Identify discontinuities and singularities through the category of the function or operations involved. These will show up as breaks in the graph. Describe the function’s behavior around discontinuities and singularities with limit notation.

\(\blacktriangleright \) End-Behavior
Think in terms of very very very big positive and negative numbers. How does the function settle down for large domain numbers? Describe end-behavior with limit notation.

Horizontal asymptotes are clues to end-behavior. Use technology to help you visualize.

\(\blacktriangleright \) Behavior

Rate-of-Change: Identify intervals where the function increases and decreases will help identify extreme values, which often occur at critical numbers. If you have the derivative, then its sign can tell you about the behavior of the original function.

Extreme Values: Generally speaking, obtaining exact values of local maximums and mimimums and their critical numbers is going to be difficult without Calculus. However, we know a lot about these in special situations.

If you have the derivative, then you can get exact values for critical numbers, which can be found where the derivative is zero or undefined.

\(\blacktriangleright \) Range
Generally speaking, you need all of the above informaiton to determine the range of a function. A graph can be very helpful.

We want exact values when we can get them. Otherwise, we settle for approximations.

Complete Anlaysis

Completely analyze \(p(t) = (t+3)e^{-2t-3}\).

\(p(t)\) is the product of a polynomial and an exponential function. Therefore, its implied domain is all real numbers.

\(p(t)\) is in factored form. The exponential factor, \(e^{-2t-3}\), does not have a zero, since it is exponential. The linear factor has a zero when \(t+3=0\) or \(t=\answer {-3}\).

The exponential factor, \(e^{-2t-3}\), only has positive values. Since the exponential factor only has positive values, the sign of \(p(t)\) is the same as the sign of \(t+3\). Therefore, \(p(t)\) is positive on \(\left ( \answer {-3}, \infty \right )\) and negative on \(\left ( -\infty , \answer {-3} \right )\).

As for end-behavior, this exponential factor dominates over the polynomial factor. The exponential values tend to \(0\) when the exponent takes on large negative values. That would be when \(t\) tends to \(\infty \). The exponential values tend to \(\infty \) when the exponent takes on large positive values. That would be when \(t\) tends to \(-\infty \).

This tells us that

• \(p(t)\) tends to \(0\) from the positive side as \(t \to \infty \).
• \(p(t)\) tends to \(-\infty \) as \(t \to -\infty \).

Neither polynomials nor exponentials have discontinuities or singularities. Therfore, \(p(t)\) is continuous on \((-\infty , \infty )\).

We can begin to piece together a sketch of the graph.

The graph suggests that there is a hill between \(-4\) and \(0\), which means there is a global maximum somewhere on \((-4, 0)\). Without a derivative, we cannot pin this critical number down. So, we’ll have to approximate. But we now know enough to make a nice sketch of the function.

There is no vertical asymptote. Exponential functions do not have singularities and neither do polynomials. The domain is \((-\infty , \infty )\) and the graph continues down and to the left.

\(y = 0\) is a horizontal asymptote.

From the graph we can estimate the critical number as approximately \(-2.5\).

• \(p(t)\) increases on \((-\infty , -2.5]\).
• \(p(t)\) decreases on \([-2.5, \infty )\).

\(p(t)\) has a global maximum of approximately \(3.7\) at \(-2.5\).

\(\lim \limits _{t \to -\infty } p(t) = -\infty \)

\(\lim \limits _{t \to \infty } p(t) = 0\)

If we are given the derivative, we could have determined the critical number and maximum value.

\(p'(t) = -(2t+5) e^{-2t-3}\), which equals \(0\) at \(-2.5\). Our estimation was right on target.

The maximum value is \(p(-2.5) = (-2.5+3)e^{-2(-2.5)-3} = 0.5 \, e^{2}\). This is approximately \(3.6945\).