Pregunta
Given:
10.1 Write down the equations of the asymptotes of
.
10.2 Determine coordinates of B , the
-intercept of
.
10.3 Write down the domain of
if
.
10.4 One of the axes of symmetry of
is an increasing function. Write down the equation
of this axis of symmetry.
10.1 Write down the equations of the asymptotes of
10.2 Determine coordinates of B , the
10.3 Write down the domain of
10.4 One of the axes of symmetry of
of this axis of symmetry.
Ask by Mills Chang. in South Africa
Mar 15,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
10.1 Asymptotes
- Vertical:
- Horizontal:
10.2
-intercept
10.3 Domain of
All real numbers except
All real numbers except
10.4 Axis of symmetry
Solución
10.1 Asymptotes
The function is
- Vertical asymptote: Occurs when the denominator is zero. Set
- Horizontal asymptote: As
,
Thus, the asymptotes are:
- Vertical:
. - Horizontal:
.
10.2
-intercept
The
-intercept is found by setting
:
Subtract 1 from both sides:
Multiply both sides by
(noting that
):
This gives:
Solve for
:
The coordinates of the
-intercept
are:
10.3 Domain of
where
We have:
The denominator
cannot be zero:
Thus, the domain of
is:
10.4 Axis of symmetry
The function
can be viewed as a hyperbola with center at the intersection of its asymptotes. The asymptotes are
and
, so the center is at:
A hyperbola with center
is symmetric about both the lines
We are asked for the axis of symmetry that is an increasing function (slope
). This line is given by:
Solving:
Thus, the axis of symmetry is:
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Mind Expander
The function
has a vertical asymptote where the denominator is zero, which occurs at
. The horizontal asymptote is determined by the behavior of the function as
approaches infinity (or negative infinity). Here, as
,
approaches
. Therefore, the asymptotes are
(vertical) and
(horizontal).
To find the
-intercept of the function, we set
:
This means the
-intercept, point B, has coordinates
.
Now, consider the transformation function
. The vertical asymptote moves to
, so the domain of
is all real numbers except for
, or in interval notation,
.
The function
is not symmetrical about the y-axis but has an axis of symmetry that is an increasing function, represented by the line through the point at which
reflects over itself. The configurations of the problem imply the axis of symmetry could be expressed as
or in a simpler linear form where
. In this setting, we want to find corresponding values for constants that make the function symmetrical without a specific numerical solution readily solvable, but typically will fall back to
for a basic orientation around the origin in such isomorphic mappings.

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