Okay, so I think these are right, but I would appreciate if someone could check them and tell me if something is wrong and what the right answer is. I'd also appreciate an explanation if possible. :)Thank you!

7. Given that
f(x)={x^3 if x ≥ 0
{x if x < 0

which of the following functions is even?

I. f(x)
II. f(|x|)
III. |f(x)|

I only
***II only
I and II only
I and III only
None of these

8. If f(x) is an odd function, which of the following must also be odd?

***–f(x)
f(|x|)
|f(x)|
f(x – 1)
None of these

9. If f(x) is an odd function, which of the following must be even?

-f(x)
f(|x - 1|)
| f(x)|
***f(x + 1)
None of these

10. If f(x) is an even function and g(x) is an odd function, which of the following must be even?
I. f(g(x))
II. f(x) + g(x)
III. f(x)g(x)

I only
II only
I and II only
II and III only
***I, II, and III

11. If f and g are odd functions, which of the following must also be odd?
I. f(g(x))

II. f(x) + g(x)

III. f(x)g(x)


I only
II only
I and II only
II and III only
I, II, and III

no answer after 7 years wow.

Oh, for 11 I chose A (first option)

#7. None of these.

Plug in -x for x and see that the graphs are not symmetric about the origin. I mean really -- part of the graph is a straight line, and part is a cubic curve.

#8 ok

#9 f(x) is odd, so f(-x) = -f(x)
|f(-x)| = |-f(x)| = |f(x)|
so, even

f(x+1) is the same graph shifted left by 1, so it is no longer symmetric about the origin.

#10
Please use a little common sense.
Pick f(x) = x^2
g(x) = x
f+g = x^2+x, neither even nor odd
fg = x^3, odd
so, we are left with
f(g(-x)) = f(-g(x)) = f(g(x)), even

#11
f(-x)+g(-x) = -f(x)-g(x) = -(f+g), so odd
f(g(-x)) = f(-g(x)) = -f(g(x)), so odd
f(-x)g(-x) = -f(x) * -g(x) = f*g, so even
Answer is I and II only

Bad logic above:

#7. Graph is part line and part curve, so no axis of symmetry

#9. f is odd, so f(x+1) is just an odd function shifted left. Still no axis of symmetry.

7. To determine which of the given functions is even, we need to apply the definition of an even function. An even function satisfies the property f(x) = f(-x) for all x in the function's domain.

I. f(x): This function is not necessarily even because it has different rules for positive (x^3) and negative (x) values of x.

II. f(|x|): To check if this function is even, we substitute -x for x and simplify: f(|-x|) = f(x). Since f(x) = f(|x|), this function is even.

III. |f(x)|: To check if this function is even, we substitute -x for x and simplify: |f(-x)| = |f(x)|. Since |f(x)| is equal regardless of the sign of x, this function is even.

Based on the analysis above, the correct answer is (II and III), so the answer is: I and III only.

8. To determine which of the given functions must be odd if f(x) is an odd function, we apply the definition of an odd function. An odd function satisfies the property f(x) = -f(-x) for all x in the function's domain.

Among the options provided, -(f(x)) satisfies the odd function definition because -(f(x)) = -(-f(-x)) = f(-x). Therefore, the correct answer is: -f(x).

9. To determine which of the given functions must be even if f(x) is an odd function, we apply the definition of an odd function. An odd function satisfies the property f(x) = -f(-x) for all x in the function's domain.

Among the options provided, f(x + 1) satisfies the even function definition because f(x + 1) = -f(-(x + 1)) = -f(-x - 1) = -(-f(x - 1)) = f(x - 1). Therefore, the correct answer is: f(x + 1).

10. If f(x) is an even function and g(x) is an odd function, we need to determine which of the given expressions must be even.

I. f(g(x)): To determine the evenness of f(g(x)), we substitute -x for x in f(g(x)) and simplify: f(g(-x)) = f(-g(x)). Since f(x) is even, f(-g(x)) = f(g(x)), making this expression even.

II. f(x) + g(x): To determine the evenness of f(x) + g(x), we substitute -x for x in f(x) + g(x) and simplify: f(-x) + g(-x) = f(x) - g(x). Since f(x) is even and g(x) is odd, f(x) - g(x) is an expression that can be both even or odd, depending on the specific functions f(x) and g(x).

III. f(x)g(x): To determine the evenness of f(x)g(x), we substitute -x for x in f(x)g(x) and simplify: f(-x)g(-x) = f(x)g(x). Since f(x) is even and g(x) is odd, f(x)g(x) is an expression that can be both even or odd, depending on the specific functions f(x) and g(x).

Based on the analysis above, the correct answer is: I, II, and III.

11. If f and g are odd functions, we need to determine which of the given expressions must also be odd.

I. f(g(x)): Since f and g are odd functions, f(g(x)) = -f(-g(x)). Combining the odd properties, we have -f(-g(x)) = f(g(x)), making this expression odd.

II. f(x) + g(x): Since f and g are odd functions, f(x) + g(x) = -f(-x) + g(-x). Combining the odd properties, we have -f(-x) + g(-x) = f(x) - g(x), which can be both even or odd depending on the specific functions f(x) and g(x).

III. f(x)g(x): Since f and g are odd functions, f(x)g(x) = -f(-x)g(-x). Combining the odd properties, we have -f(-x)g(-x) = f(x)g(x), making this expression odd.

Based on the analysis above, the correct answer is: I only.