We are given the function g(x) = - 10 x^4 - 90
And we are asked to find which statement is not true.
First statement reads:
Its graph has a y-intercept at (0,-90)
This statement is correct, since when we set x = 0 in the function, we get:
g(0) = - 10 *0^4 - 90 = 0 - 90 = -90
Second statement reads:
The function has at least ONE real root.
This statement is NOT true, since there is no value of x for which the function is zero (that meaning the function doesn't cross the x axis.
We can verify such requesting that g(x) = 0 = - 10 x^4 - 90
adding 10 x^4 to both sides:
10 x^4 = - 90
dividing both sides by 10 to isolate x^4
x^4 = - 90 / 10
x^4 = - 9
we realize at this point that there is no x for which the function is equal to zero (so there is no root) since we cannot find any value of x that raised to an EVEN power can give a negative number.
This second statement is therefore NOT TRUE, and the statement we need to mark.
The other two statements are true:
Third statement that reads
as x approaches negative infinity, g(x) approaches negative infinity.
We can check that being true using very negative values of x as for example -100. when we use such, the function becomes:
g(-100) = - 10 (-100)^4 - 90 = - 1000000000 - 90 which as we see is a very negative number ( the functionreally approaches negative infinity for very negative x values)
Similarly, the fourth statement is also true, since when we use large values for x (for example 100) we again get a veru negative value that indicates that the function is approaching negative infinity as well.
g(100) = - 10 (100)^4 - 90 = - 1000000000 - 90