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h(x) = 1/x - 4, F(x) = 1/x, m(x) = -8/x + 2

July 29, 2026Mathematics

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Step-by-step solution

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Here is an analysis of the given functions:

Function 1: $h(x) = \frac{1}{x} - 4$

Identify the type of function and its asymptotes.
This is a hyperbola.
The vertical asymptote occurs where the denominator is zero.
$x = 0$
The horizontal asymptote is determined by the constant term.
$y = -4$
This function is a vertical translation of the parent function $F(x) = \frac{1}{x}$ downwards by 4 units.

Function 2: $F(x) = \frac{1}{x}$
Identify the type of function and its asymptotes.
This is the parent hyperbola.
The vertical asymptote occurs where the denominator is zero.
$x = 0$
The horizontal asymptote is the constant term, which is 0 in this case.
$y = 0$
Function 3: $m(x) = -\frac{8}{x} + 2$
Identify the type of function and its asymptotes.
This is a hyperbola.
The vertical asymptote occurs where the denominator is zero.
$x = 0$
The horizontal asymptote is determined by the constant term.
$y = 2$
This function is a vertical stretch by a factor of 8, reflected across the x-axis, and then translated upwards by 2 units from the parent function $F(x) = \frac{1}{x}$ .

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