Question

Use the graph to determine
  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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x y graph
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Part 1
a. Enter the​ domain:
  
y ​(Type your answer in interval​ notation.)

Answer 1

To determine the properties of the function \( f(x) \) based on the given description (and assuming a general shape for the graph), we can analyze the function as follows:

### Part 1: Domain
The domain of a function is the set of all possible input values (x-values). Typically, for many functions, unless specified otherwise, the domain can be all real numbers. However, if \( f(x) \) approaches but never touches the x-axis, it could indicate a horizontal asymptote at \( y = 0 \) but allows \( x \) to take on any real number.

Therefore, assuming no restrictions on \( x \):

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

### Part 2: Range
The range of a function is the set of all possible output values (y-values). Since you mention that the function approaches but never touches the x-axis, it suggests that the values continue indefinitely towards the y-axis.

If we assume the function never goes below the x-axis (y = 0) and can take positive values, e.g., if it is an exponential function or similar, the range would be:

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

### Part 3: X-intercepts
X-intercepts are points where the graph of the function crosses the x-axis (where \( y = 0 \)). Given the information that the function approaches but never touches the x-axis, we can conclude that:

**X-intercepts: none**

### Part 4: Y-intercept
The y-intercept is the point where the graph crosses the y-axis (where \( x = 0 \)). To determine the y-intercept, we generally evaluate \( f(0) \). As we do not have a specific function but based on your description, let's assume \( f(0) \) results in a positive value.

**Y-intercept: \( f(0) = k > 0 \) (assuming \( k \) is some positive constant)**

If you could provide the actual numerical y-axis or specific examples or values for the function, I could refine this information further. However, from your description and typical behavior of functions, the above conclusions hold.