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Graphing rational functions
1

Graphing rational functions 1-example-1

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6 votes

Answer:

(76+92x=4)

Explanation:

User Ralph King
by
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3 votes

The horizontal asymptote is at y=4, shown by the green dashed line. This indicates the behavior of the function as x becomes very large in magnitude.

The blue dot marks the y-intercept at (0,6).

To find the graph that represents the function
\( f(x) = (2)/(x-1) + 4 \), let's analyze its characteristics step by step:

1. Identify the Type of Function

The function
\( f(x) = (2)/(x-1) + 4 \) is a rational function, specifically a transformed version of the reciprocal function.

2. Vertical Asymptote

The denominator
\( x - 1 = 0 \) gives the vertical asymptote, which is
\( x = 1 \). This means the graph will approach but never touch the line
\( x = 1 \).

3. Horizontal Asymptote

For large values of x, the term
\( (2)/(x-1) \) approaches zero. Thus, the horizontal asymptote is determined by the constant term, which is
\( y = 4 \).

4. Y-Intercept

To find the y-intercept, set
\( x = 0 \):


\[ f(0) = (2)/(0-1) + 4 = 2 + 4 = 6 \]

So, the y-intercept is at (0, 6).

5. Behavior around Asymptotes

  • As x approaches 1 from the left,
    \( f(x) \) goes to negative infinity.
  • As x approaches 1 from the right,
    \( f(x) \) goes to positive infinity.

6. Additional Points

It might be helpful to calculate the function's value at a couple of additional points for a more accurate graph. For instance, calculate
\( f(2) \) and
\( f(-1) \).

7. Sketching the Graph

Considering these characteristics, the graph:

  • Has a vertical asymptote at
    \( x = 1 \).
  • Has a horizontal asymptote at
    \( y = 4 \).
  • Passes through the y-intercept at (0, 6).
  • Shows the function going to negative infinity as x approaches 1 from the left and to positive infinity as x approaches 1 from the right.

Here is the graph of the function
\( f(x) = (2)/(x-1) + 4 \). As analyzed:

(Graph is give below)

It has a vertical asymptote at x=1, represented by the red dashed line. The function goes to negative infinity as x approaches 1 from the left and to positive infinity as x approaches 1 from the right.

The complete question is here:

Which graph represents the function
$f(x)=(2)/(x-1)+4$ ?

Graphing rational functions 1-example-1
User Dipiks
by
8.4k points

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