consider the functions g(x)=(4-x^2)^0.5 and h(x)=(x^2)-5.
find the composite functions hog(x) and goh(x), if they exist, and state the domain and range of the composite functions.
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ok, i have figured out that hog(x)=-1-X^2, but in my answer booklet it says that the domain of this function is -2<&=x<&=2, and the range is -5<&=y<&=-1. can someone please explain how i can get this answer??thanks
and i have no idea why the function goh(x) does not exist. help please!
hog(x) means that you first perform the function g(x), and then enter the result of that function as x in the second function h(x).
The function g(x) is a square root, meaning that whatever is under the square root sign has to be positive (you can't take the square root of a negative number).
4-x^2 is only positive if -2 <= x <= 2, so this is the domain of g(x) and since x can take on any value in h(x), it's also the domain of hog(x).
Now, for the range, we have to check when h(x) takes on its largest and smallest values. When we draw the graph of x²-5, we notice that it is a parabole. We can easily determine that for x=0, the parabole will reach its lowest point, and then starts to rise again. Since the maximum value for x is 2 or -2, we check that for 2 or -2 h(x) reaches the value -1 (being its largest value for -2 <=x <= 2).
So the range of hog(x) is -5<=y<=-1.
As for goh(x), it does exist. if we replace every x in g(x) by h(x), we get goh(x). Since 4-(x²-5)² is only positive for -sqrt(7) until -sqrt(3) and from sqrt(3) untill sqrt(7), this function only has values in those places.
So the domain of goh(x) = [-sqrt(7),-sqrt(3)] U [sqrt(3),sqrt(7)].
write h(x) as composition of two functions h(x) = (fog)(x)
h(x) = 3 square root 7 + 3 square root of x
To find the composite functions hog(x) and goh(x), we need to substitute one function into the other.
First, let's find hog(x):
hog(x) = h(g(x))
= h((4-x^2)^0.5)
= ((4-x^2)^0.5)^2 - 5
= 4-x^2 - 5
= -x^2 - 1
So, hog(x) = -x^2 - 1.
The domain of hog(x) is the same as the domain of g(x), which is determined by the square root function. In this case, (4-x^2) must be greater than or equal to 0:
4 - x^2 ≥ 0
x^2 ≤ 4
-2 ≤ x ≤ 2
Therefore, the domain of hog(x) is -2 ≤ x ≤ 2.
To find the range of hog(x), we need to consider the range of g(x) and the range of h(x). The range of g(x) is determined by the square root function and can be any non-negative real number. However, since h(x) = x^2 - 5, the range of h(x) is all real numbers greater than or equal to -5.
Combining the ranges of g(x) and h(x), the range of hog(x) is -5 ≤ y ≤ -1.
Now let's consider goh(x):
goh(x) = g(h(x))
= g(x^2 - 5)
= (4 - (x^2 - 5))^0.5
= (9 - x^2)^0.5
To determine the existence of goh(x), we need the expression inside the square root, 9 - x^2, to be greater than or equal to 0:
9 - x^2 ≥ 0
x^2 ≤ 9
-3 ≤ x ≤ 3
Since the range of h(x) includes all real numbers greater than or equal to -5, when we substitute h(x) into g(x), the expression becomes (9 - (h(x)))^0.5. However, for values of x outside the range of h(x), the expression (9 - x^2)^0.5 will not exist. In this case, for x > 3 or x < -3, the composite function goh(x) does not exist.
In summary:
- hog(x) = -x^2 - 1, with domain -2 ≤ x ≤ 2 and range -5 ≤ y ≤ -1.
- goh(x) does not exist.
To find the composite function hog(x), we need to substitute h(x) into g(x).
Starting with h(x) = x^2 - 5, we substitute this into g(x) = (4 - x^2)^0.5. This gives us:
hog(x) = g(h(x)) = (4 - (x^2 - 5))^0.5 = (9 - x^2)^0.5
To simplify this further, we can rewrite (9 - x^2)^0.5 as (-1)(x^2 - 9)^0.5, since the square root of a negative number is imaginary. Additionally, we can rewrite (x^2 - 9)^0.5 as (-(x^2 - 9))^0.5, since the square root of a positive number can be positive or negative. So, we have:
hog(x) = (-1)(-(x^2 - 9))^0.5 = -((x^2 - 9)^0.5)
Thus, hog(x) = -((x^2 - 9)^0.5) = -((x - 3)(x + 3))^0.5 = -√(x - 3)(x + 3) = -√(x - 3)√(x + 3)
Now let's analyze the domain and range of hog(x):
Domain: The original domain of h(x) is all real numbers, since there are no restrictions on x in h(x) = x^2 - 5. However, to have a real value for hog(x), we need (x - 3)(x + 3) to be greater than or equal to 0, since we can't take the square root of a negative number. By factoring (x - 3)(x + 3), we find that it is positive when -3 < x < 3. Therefore, the domain of hog(x) is -3 < x < 3.
Range: To find the range of hog(x), we can analyze the expression √(x - 3)√(x + 3). The square root of a number is always positive or zero, so √(x - 3) and √(x + 3) are both greater than or equal to 0. When multiplied together, they will result in a non-positive or zero value since one is negative and the other is non-negative. Therefore, the range of hog(x) is -∞ ≤ y ≤ 0.
Now let's discuss why the function goh(x) does not exist.
For goh(x) to exist, we need to substitute g(x) into h(x), but we can't do that because h(x) = x^2 - 5 is not defined for non-real values of x, including imaginary numbers. Since g(x) = (4 - x^2)^0.5 would result in non-real values for x^2 > 4, substituting g(x) into h(x) is not possible.
Hence, goh(x) does not exist.