Question
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The function is defined as follows.
(a) Find the domain of the function.
(b) Locate any intercepts.
© Graph the function.
(d) Based on the graph, find the range.

Ask by Smith Floyd. in the United States
Nov 04,2024

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Tutor-Verified Answer

Answer

  • Domain:
    • Y-intercept:
    • X-intercept: None
    • Range:

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Mind Expander

(a) The function is defined piecewise for all real numbers . Since there are no restrictions on the values that can take, the domain of the function is .
(b) To find the intercepts, we look for the y-intercept first by setting :
Next, for the x-intercepts, we set :
  1. For :
  2. For :
Thus, the function has a y-intercept at but no x-intercepts.
© To graph , we plot the two pieces:
  • For , the equation is , which is a line with a slope of -2 and y-intercept at . The line extends to the left until it approaches the point at but does not include it because it is exclusive.
  • For , the equation is , which starts at and continues to the right with a slope of 2.
Combining these pieces visually shows a break in the graph at and indicates different slopes on either side.
(d) To find the range, observe the outputs of each piece of the function:
  • For (from to ), the outputs go from as approaches down to .
  • For , as increases, yields values starting from at and upwards towards .
Thus, the range of the function is .

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