Determine the range of each function
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
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
I suppose that "XER" means the domain (all possible values of x) is all real numbers. And the answers there are both...wrong.
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
To determine the range of a function, you need to find the set of all possible output values of the function for all possible input values.
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.