Chapter Review
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If a horizontal line can intersect the curve at more than one point, then the inverse is not a function. To find the inverse, start by replacing f(x) with y.
Is the inverse a function? Yes
Inverse of the function: g(x)= 3x-6
For the given function, we will first draw its graph and then use the Horizontal Line Test to determine whether the inverse is a function. Then, we will find the inverse. Let's start!
The above means that x= -6 is the vertical asymptote of the function. To find the horizontal asymptote, let's pay close attention to the degrees of the numerator and denominator. f(x)=3/x^1+6 We see that the degree of the denominator is greater than the degree of the numerator. Therefore, the line y=0 is a horizontal asymptote. Now, we will draw the asymptotes.
Next, we will make a table of values. We will include x-values to the left and to the right of the vertical asymptote.
| x | 3/x+6 | f(x)=3/x+6 |
|---|---|---|
| - 10 | 3/- 10+6 | - 3/4 |
| - 9 | 3/- 9+6 | - 1 |
| - 8 | 3/- 8+6 | - 3/2 |
| - 7 | 3/- 7+6 | -3 |
| -5 | 3/-5+6 | 3 |
| -4 | 3/-4+6 | 3/2 |
| -3 | 3/-3+6 | 1 |
| -2 | 3/-2+6 | 3/4 |
Let's plot and connect the points.
Finally, we can perform the Horizontal Line Test. If the horizontal lines intersect the graph once, then the inverse is also a function. Conversely, if there is even one horizontal line that intersects the graph more than once, then the inverse is not a function.
We can see above that all the horizontal lines intersect the curve at only one point. Therefore, the inverse of the given function is also a function.
LHS * (y+6)=RHS* (y+6)
a* b/c=a*b/c
a/a=1
.LHS /x.=.RHS /x.
LHS-6=RHS-6