Use the graph to determine

Question

  a. the​ function's domain;
  b. the​ function's range;
  c. the​ x-intercepts, if​ any;
  d. the​ y-intercept, if any.
The​ function, f(x) approaches but never touches the​ x-axis.
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Part 1
a. Enter the​ domain:
  
enter your response here ​(Type your answer in interval​ notation.)

Answer 1

To determine the characteristics of the function \( f(x) \) based on the provided information, let's analyze each part:

a. The function's domain

The domain of a function is the set of all possible input values (x-values) that the function can accept. Based on the typical graph behavior you've described, which approaches but never touches the x-axis, we can infer that it could potentially take all real numbers.

If there are no restrictions or vertical asymptotes noted in your description, we can conclude:

Domain: \( (-\infty, \infty) \)

b. The function's range

The range of a function is the set of possible output values (y-values). Since the function approaches the x-axis but never touches it, it is likely that the function's values are either strictly positive or strictly negative, depending on its position relative to the x-axis.

If we assume \( f(x) \) approaches the x-axis from above (meaning the function approaches \( y = 0 \) but does not touch it):

Range: \( (0, \infty) \)

If it approaches from below, then:

Range: \( (-\infty, 0) \)

Please clarify if you have any specific information about the function's behavior around the x-axis to finalize this.

c. The x-intercepts, if any

The x-intercepts are the values of \( x \) for which \( f(x) = 0 \). Since the function approaches the x-axis but never touches it, it means it has no points where it intersects the x-axis.

X-intercepts: None

d. The y-intercept, if any

The y-intercept occurs where the graph intersects the y-axis (at \( x = 0 \)). To find this, one would typically evaluate \( f(0) \). In this case, without further information about the function's specific behavior, you would need to look at the graph to determine its value at \( x = 0 \).

If we assume the function approaches a particular value as it approaches the y-axis (or based on the graph), you would provide that value. If you give no specific mention of a function value or graphical indication around \( x = 0 \):

Y-intercept: Might be \( f(0) = \) a specific value (you'll need to provide details or reference to the graph)

If none appears to be provided, state whether the y-intercept exists based on the graph.

Please provide additional details or references for part d to clarify the y-intercept.