If a horizontal line can intersect the curve at more than one point, then the inverse is not a function.
No.
Practice makes perfect
We can use the Horizontal Line Test to determine whether the inverse of a function is also a function. To do this with the given function, we will first make a table of values to graph it. When making a table of values, make sure to use a variety of points, including negative and positive values.
x
4x^3-5x
g(x)
- 3
4( - 3)^3-5( - 3)
- 93
- 2
4( - 2)^3-5( - 2)
- 22
- 1
4( - 1)^3-5( - 1)
1
0
4( 0)^3-5( 0)
0
1
4( 1)^3-5( 1)
- 1
2
4( 2)^3-5( 2)
22
3
4( 3)^3-5( 3)
93
Now we can plot the obtained points and connect them with a smooth curve. Consider also that this is an odd-degreepolynomial with a positive leading coefficient. This tells us about the end behavior of the function.
&g(x) → - ∞ as x → - ∞
&g(x) → + ∞ as x → + ∞
Let's draw the function!
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 there is at least one horizontal line that intersect the curve at more than one point. Therefore, the inverse of the given function is not a function.
