Did I get this practice question right?
1. Consider the following functions:
f(x)=cos(x^3-x)
h(x)=|x-3|^3
g(x)=1n(|x|+3)
s(x)=sin^3(x)
Which of the following is true?
(Points : 1)
f is even, h and s are odd.
***f and g are even, s is odd.
h and s are odd, g is even.
s is odd, f and h are even.
g and f are even, h is odd.
correct
Well, if "h" is odd, it probably needs a psychiatrist. As for "g," it's always nice to have a friend who is "even" keeled. And "s" seems like the odd one out. So, the correct answer is: f and g are even, s is odd. Keep up the good work!
To determine which of the following is true, we need to analyze the given functions.
f(x) = cos(x^3 - x)
g(x) = ln(|x| + 3)
h(x) = |x - 3|^3
s(x) = sin^3(x)
To determine whether a function is even or odd, we need to analyze their symmetries.
1. f(x) = cos(x^3 - x)
The function f(x) is not symmetric with respect to the y-axis (not even) since it involves the non-symmetric term x^3. To check if it is odd, we need to check if f(-x) = -f(x).
f(-x) = cos((-x)^3 - (-x))
= cos(-x^3 + x)
= cos(x^3 - x)
Since f(-x) = f(x), f(x) is not odd.
2. g(x) = ln(|x| + 3)
The function g(x) involves the natural logarithm, which is not defined for negative values of x. Hence, the domain of g(x) is x >= -3. Therefore, g(x) is not an even function.
3. h(x) = |x - 3|^3
The absolute value function |x - 3| is symmetric with respect to the y-axis (even), and raising it to the power of 3 preserves this symmetry. Therefore, h(x) is an even function.
4. s(x) = sin^3(x)
The function s(x) involves the sine function, which is not symmetric with respect to the y-axis (not even). To check if it is odd, we need to check if s(-x) = -s(x).
s(-x) = sin^3(-x)
= (-sin(x))^3
= -sin^3(x)
Since s(-x) = -s(x), s(x) is an odd function.
From our analysis, we can conclude that f and g are even functions, and s is an odd function. Thus, the correct answer is "f and g are even, s is odd."
To determine if you got the practice question right, let's analyze the functions given and evaluate their properties.
1. f(x) = cos(x^3 - x)
- This function involves the cosine of (x^3 - x).
- To check if it's even or odd, we evaluate f(-x) and see if it's equal to f(x).
- f(-x) = cos((-x)^3 - (-x)) = cos(-x^3 + x) = cos(x^3 - x) = f(x)
- f(x) = f(-x), so f(x) is an even function.
2. h(x) = |x - 3|^3
- This function involves the absolute value of (x - 3) raised to the power of 3.
- To check if it's even or odd, we evaluate h(-x) and see if it's equal to -h(x).
- h(-x) = |(-x) - 3|^3 = |-(x + 3)|^3 = |x + 3|^3
- -h(x) = -|x - 3|^3 = -|x + 3|^3
- h(-x) = -h(x), so h(x) is an odd function.
3. g(x) = ln(|x| + 3)
- This function involves the natural logarithm of (|x| + 3).
- To check if it's even or odd, we evaluate g(-x) and see if it's equal to g(x).
- g(-x) = ln(|-x| + 3) = ln(|x| + 3) = g(x)
- g(x) = g(-x), so g(x) is an even function.
4. s(x) = sin^3(x)
- This function involves the sine of x raised to the power of 3.
- To check if it's even or odd, we evaluate s(-x) and see if it's equal to -s(x).
- s(-x) = sin^3(-x) = (-sin(x))^3 = -sin^3(x) = -s(x)
- s(-x) = -s(x), so s(x) is an odd function.
Based on the analysis, f and g are even functions, and s is an odd function. Therefore, the correct answer is: f and g are even, s is odd.