x^2+3 >= 3 for all x
so, f(x) >= 9
3+x^4 >= 3 for all x
1/(3+x^4) <= 1/3 for all x
as x gets huge, 1/x^4 -> 0 so,
0 < f(x) <= 1/3
wolframalpha.com will graph them for you to see
a)f(x)=(x^2+3)^2,XER
Answer is f(x)ER, f(x)>9
b)f(x)=1/3+x^4,XER
Answer is f(x)ER, 0<f(x)<1/3
Please explain how they go those two answers
so, f(x) >= 9
3+x^4 >= 3 for all x
1/(3+x^4) <= 1/3 for all x
as x gets huge, 1/x^4 -> 0 so,
0 < f(x) <= 1/3
wolframalpha.com will graph them for you to see
Anyway, range is the set of all possible values of y (or f(x)). Let's analyze each problem. :)
a) f(x) = (x^2+3)^2
Note that x^2 can never be negative, because any real number squared is always greater than or equal to zero. Thus the smallest possible value of x^2 is zero, and if we substitute it in f(x),
f(x) = (0^2 + 3)^2
f(x) = 3^2
f(x) = 9, which is the smallest possible value of f(x)
Thus the range is all real numbers greater than or equal to 9, or f(x) >= 9
b) f(x) = 1/3+x^4
x^4 is also (x^2)^2. Then again, x^4 can never be negative. So the smallest possible value of x^4 is zero, and substituting,
f(x) = (1/3) + 0^4
f(x) = 1/3
Thus the range is all real numbers greater than or equal to 1/3, or f(x) >= 1/3
Hope this helps~ :3
a) For the function f(x) = (x^2 + 3)^2, where x belongs to the set of real numbers (XER):
To find the range, first, consider that the expression (x^2 + 3)^2 will always yield a non-negative result since squaring any real number will result in a non-negative value.
So, f(x) will always be greater than or equal to zero because the square of any real number is non-negative.
To find the minimum value of f(x), we need to find the minimum possible value of (x^2 + 3)^2. Since square numbers are always non-negative, the minimum possible value is when the expression evaluates to zero. Thus f(x) = 0 when x = 0.
Therefore, the range of f(x) is any non-negative real number: f(x)ER, where f(x) > 0.
Additionally, since f(x) = (x^2 + 3)^2, and (x^2 + 3) will always be greater than or equal to 3, we can conclude that f(x) > 9 for any xER.
So, the full range of the function is f(x)ER, f(x) > 9.
b) For the function f(x) = 1/3 + x^4, where x belongs to the set of real numbers (XER):
To find the range, we need to determine the possible values of f(x).
The term x^4 is always non-negative because a positive number raised to an even power is always non-negative.
So, the range of x^4 is [0, +∞).
Now, since we are adding 1/3 to the values of x^4, the minimum possible value will be when x^4 = 0, resulting in f(x) = 1/3.
Therefore, the minimum value of f(x) is 1/3, and it can increase indefinitely as x^4 increases.
Since x^4 can vary between 0 and +∞, the range of f(x) is 0 < f(x) < 1/3.
So, the full range of the given function is f(x)ER, 0 < f(x) < 1/3.