Question

Consider function f.



Which statement is true about function f?

A.
The function is continuous.
B.
As x approaches positive infinity, approaches positive infinity.
C.
The function is increasing over its entire domain.
D.
The domain is all real numbers.

Answer 1

Answer:

Step-by-step explanation:

Check for continuity by evaluating 2^x and -x^2 - 4x + 1 at the break point x = 0:  2^0 is 1 and -x^2 - 4x + 1 is also 1, so these two functions approach the same value as x approaches 0.

Now do the same thing with

-x^2 - 4x + 1 and (1/2)x + 3 at x = 2; the first comes out to -11 and the second to 4.  Thus, this function is not continuous at x = 2.  

We must reject statement A.

Statement B:  as x increases without bound, (1/2)x + 3 also increases without bound.  This statement is true.

Statement C:  False, because the quadratic -x^2 - 4x + 1 has a maximum at

x = -b/[2a], or x = -(-4)/[-2], or x = -2

Statement D:  True:  there are no limitations on the values of the input, x.

Answer 2

The statement B is as x approaches positive infinity f(x) approaches positive infinity is true

What is the definition of the limit?

A point or level beyond which something does not or may not extend or pass.

We have to check for continuity by evaluating $2^x$ and $-x^2 - 4x + 1$

at the break point x = 0

$2^0$is 1 and $-x^2 - 4x + 1$ is also 1,

So these two functions approach the same value as x approaches 0.

Now do the same thing with

$-x^2 - 4x + 1$and $(1/2)x + 3$ at x = 2;

The first comes out to -11 and the second to 4.  

Thus, this function is not continuous at x = 2.  

We must reject statement A.

for B we have as x increases without bound,

$(1/2)x + 3$

also increases without bound.  

Therefore the statement B is true statement.

Statement C is  False,

because the quadratic$-x^2 - 4x + 1$ has a maximum at

$x = -b/[2a],$

$x = -(-4)/[-2],$

$x=-2$

There are no limitations on the values of the input, x.

D is also false negative number are in real numbers.

Therefore,the statement B is as x approaches positive infinity f(x) approaches positive infinity is true

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