Question

Graphing rational functions
1

Answer 1

(76+92x=4)

Explanation:

Answer 2

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
`$f(x)=(2)/(x-1)+4$` ?