Question

Which of the following equations could possibly be a formula for the graph? There may be more than one correct answer. Select all that apply. A. y = e^-x B. y = log(x) C. y = -e^-x D. y = - log(x) E. y = ln(x) F. y = - ln(x)

Answer 1

The equations that could be a formula for the given graph are A.
`\(y = e^(-x)\)`, B. y = log(x), and E. y = ln(x).

Step-by-step explanation:

The choices A, B, and E correspond to exponential and logarithmic functions that can produce specific types of graphs. Let's analyze each option:

A.
`\(y = e^(-x)\)`: This equation represents an exponential decay function. As x increases, the value
`\(e^(-x)\)` decreases exponentially, producing a curve that approaches the x-axis but never touches it.

B. y = log(x): This is the logarithmic function with base 10. The graph of log(x) increases slowly for small values of x and more rapidly for larger values, displaying logarithmic growth.

E. y = ln(x): This is the natural logarithmic function with base e. Similar to the base 10 logarithm, ln(x) increases slowly for small x and more rapidly for larger x.

The other options, C, D, and F introduce negative signs, resulting in reflections across the x-axis. These equations represent variations of the mentioned functions, but they wouldn't produce the same graph as the original functions without the negative sign. Therefore, options C, D, and F are not suitable for the given graph.

Answer 2

Without the graph, we can only describe the behavior of given functions: exponential functions (decaying or growing) and logarithmic functions (increasing or decreasing), but we cannot definitively determine which formula matches the graph.

Step-by-step explanation:

The student is trying to determine which function could represent a given graph. Without the actual graph, we can analyze each function's behavior to provide guidance. Functions y = e-x and y = -e-x are exponential functions with their growth/decay determined by the sign in front. y = log(x) and y = ln(x) are both logarithmic functions, while y = -log(x) and y = -ln(x) are their inverses with a reflection across the x-axis. In logarithmic functions, as x increases, y also increases, and a negative sign in front will reverse this behavior. Exponential functions will approach zero as x increases. Considering the behavior described in the question and provided options, we can derive some potential matches for the graph's formula. However, without an actual graph or additional details, it's impossible to provide a definitive answer.