Pregunta
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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.

Ask by Mills Chang. in South Africa
Mar 15,2025

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10.1 Asymptotes
  • Vertical:
  • Horizontal:
10.2 -intercept
10.3 Domain of
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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