If f(x) is an odd function, which function must also be odd? Explain.

(1) f(x – 1) + 5
(2) 2f(x) + 3
(3) 1/2f(x)
(4) f(x – 4)

review what it means to be odd:

f(-x) = -f(x)
now, which choice fits?

is it 4?

nope.

for #4, that is just the graph of f(x) shifted to the right by 4. So it is no longer symmetric about the origin.
It's #3.
If g(x) = 1/2 f(x) then since f(-x) = -f(x),
g(-x) = 1/2 f(-x) = 1/2 (-f(x)) = - (1/2 f(x)) = -g(x)
so g(x) is also odd

To determine which function must also be odd if f(x) is an odd function, we need to understand the properties of odd functions.

An odd function is defined as a function that satisfies the condition f(-x) = -f(x) for all x in its domain. In simpler terms, if you replace x with its opposite (-x), and the resulting function is the negative of the original function, then it is an odd function.

Let's analyze each option:

(1) f(x – 1) + 5:
To determine if this function is odd, we need to substitute (-x) for x and simplify it:
f(-x – 1) + 5 = -f(x + 1) + 5

Since the resulting function includes the negative of f(x), it is not an odd function.

(2) 2f(x) + 3:
To determine if this function is odd, we need to substitute (-x) for x and simplify it:
2f(-x) + 3

Since there is no sign change in this function, it is not an odd function.

(3) 1/2f(x):
To determine if this function is odd, we need to substitute (-x) for x and simplify it:
1/2f(-x) = -1/2f(x)

Since the sign changes occur in this function, it is an odd function.

(4) f(x – 4):
To determine if this function is odd, we need to substitute (-x) for x and simplify it:
f(-x – 4)

Since there is no sign change in this function, it is not an odd function.

Therefore, the function in option (3) - 1/2f(x) - must also be odd if f(x) is an odd function.