Decide which of the following properties apply to the function. more than one apply

Question

y = −ln x
The function is one-to-one.
The domain of the function is (−∞, ∞).
The range of the function is (−∞, ∞).
The function has a turning point.
The function is a polynomial function.
The function is increasing on its entire domain.
The graph has an asymptote.
The function is decreasing on its entire domain

Answer 1

Let's examine a few things about this function.  You cannot have a negative nor zero for an argument for logarithms.  Therefore, the domain is (0, ∞).

 

Since we know the function has a restricted domain, we also know it has an asymptote.  The vertical asymptote is x=0.

 

Lets see if the graph increases or decreases.

 

-ln(1) = 0

-ln(2.7) = -ln(e) = -1

 

As x increases, the values of y tend to slowly decrease.  However, there is no turning point because the values never change from decrease to increase.

 

Knowing these facts, can you answer the question?

Answer 2

The graph of y=-ln(x) has a vertical asymptote at x=0 (along the y-axis) and y approaches +infinity as x gets closer and closer to the y-axis.  As you move to the right, y=-ln(x) crosses the x-axis at x=1 and continues very slowly to y=-infinity as x goes out to + inifinity.    I wish I could draw it here for you, if makes all of the properties listed above easier to see.  

 

It looks similar to the left half of a "U", but it doesn't flatten out on the bottom.

 

one-to-one:  has to pass the horizontal line test.  Since only any horizontal line crosses this graph only once, they YES it is one-to-one

 

domain of the function is (-inf, +inf):  FALSE.  only positive x-values are allowed.

 

range of the function is (-inf, +inf):  TRUE   the y-values go all the way up to +inf and eventually all the way down to -infinity

 

turning point:  FALSE.  It is always decreasing, it never turns back up again.

 

polynomial:  FALSE.  It is "transcendental".  A polynomial is of the form ax^n + bx^(n-1)... +cx + k   y=-ln(x) is NOT a polynomial.

 

increasing on its entire domain:  FALSE.  It goes downhill as you follow it from left to right (same way you read English), so it is DECREASING on its entire domain.

 

graph has an asymptote:  TRUE, the y-axis (or where x=0)

 

function is decreasing on its entire domain:  TRUE.  y is +inf at far left and y is -inf at far right, and it never goes up, only farther and farther down as you move to the right.  It IS decreasing on its entire domain.

 

 

Hope that helps!

Vicki