Question
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.
(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
Upstudy AI Solution
Tutor-Verified Answer
Answer
- Domain:
- Y-intercept:
- X-intercept: None
- Range:
- Y-intercept:
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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
:
- For
: - 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
.