Suppose that the function
h
is defined, for all real numbers, as follows.
=hx
−4
<if x−1
+−+x122
≤−if 1≤x2
−14x2
>if x2
Find
h0
,
h2
, and
h5
.
To find h(0), h(2), and h(5), we can substitute the values of 0, 2, and 5 into the function h(x) and evaluate each expression step by step.
1. For h(0):
Substitute x = 0 into the function:
h(0) = (-4) if (0-1)<0^2+(-4)/2 ≤ -14(0)^2
= -4 if -1 < 0 + (-4)/2 ≤ 0
= -4 if -1 < -2 ≤ 0
= -4 if true
= -4
Therefore, h(0) = -4.
2. For h(2):
Substitute x = 2 into the function:
h(2) = (-4) if (2-1)<2^2+(-4)/2 ≤ -14(2)^2
= -4 if 1 < 4 + (-4)/2 ≤ -56
= -4 if 1 < 4 - 2 ≤ -56
= -4 if 1 < 2 ≤ -56
= -4 if true
= -4
Therefore, h(2) = -4.
3. For h(5):
Substitute x = 5 into the function:
h(5) = (-4) if (5-1)<5^2+(-4)/2 ≤ -14(5)^2
= -4 if 4 < 25 + (-4)/2 ≤ -14(25)
= -4 if 4 < 25 - 2 ≤ -350
= -4 if 4 < 23 ≤ -350
= -4 if false
= 25
Therefore, h(5) = 25.
To find the values of h(0), h(2), and h(5), we need to plug these values into the given function and evaluate the result.
For h(0):
Substitute x = 0 into the function:
h(0) = -4 if 0 - 1 ≤ 2 + 0^2
= -4 if -1 ≤ 2
= -4
So, h(0) = -4.
For h(2):
Substitute x = 2 into the function:
h(2) = -4 if 2 - 1 ≤ 2 + 2^2
= -4 if 1 ≤ 2 + 4
= -4 if 1 ≤ 6
= 0 if true
= -14(2^2) if false
= -14 * 4
= -56
So, h(2) = -56.
For h(5):
Substitute x = 5 into the function:
h(5) = -14(5^2) if 5^2 > 2
= -14 * 25 if true
= -350
So, h(5) = -350.
Therefore, h(0) = -4, h(2) = -56, and h(5) = -350.