Answer :
To determine the domain and range of the function [tex]\( f(-x) \)[/tex], we need to analyze the transformation applied to [tex]\( f(x) \)[/tex].
### Step 1: Determine the Domain of [tex]\( f(-x) \)[/tex]
The domain of [tex]\( f(x) \)[/tex] is given as [tex]\([-2, 3]\)[/tex]. When we replace [tex]\( x \)[/tex] with [tex]\(-x \)[/tex] to get [tex]\( f(-x) \)[/tex], we have to consider how this affects the domain.
- For [tex]\( f(x) \)[/tex], [tex]\( x \)[/tex] ranges from [tex]\(-2\)[/tex] to [tex]\( 3 \)[/tex].
- For [tex]\( f(-x) \)[/tex], we replace [tex]\( x \)[/tex] with [tex]\(-x \)[/tex], which means we need to reverse the sign of the bounds of the domain.
- The original domain [tex]\([-2, 3]\)[/tex] will transform to [tex]\([-(3), -(-2)]\)[/tex], which simplifies to [tex]\([-3, 2]\)[/tex].
Thus, the new domain of [tex]\( f(-x) \)[/tex] is [tex]\([-3, 2]\)[/tex].
### Step 2: Determine the Range of [tex]\( f(-x) \)[/tex]
The range of [tex]\( f(x) \)[/tex] is given as [tex]\([0, 3]\)[/tex]. Changing [tex]\( x \)[/tex] to [tex]\(-x \)[/tex] does not affect the range because the output values of the function remain the same for corresponding [tex]\( x \)[/tex] values.
Therefore, the range of [tex]\( f(-x) \)[/tex] remains [tex]\([0, 3]\)[/tex].
### Conclusion
The domain of [tex]\( f(-x) \)[/tex] is [tex]\([-3, 2]\)[/tex] and the range of [tex]\( f(-x) \)[/tex] is [tex]\([0, 3]\)[/tex].
### Step 1: Determine the Domain of [tex]\( f(-x) \)[/tex]
The domain of [tex]\( f(x) \)[/tex] is given as [tex]\([-2, 3]\)[/tex]. When we replace [tex]\( x \)[/tex] with [tex]\(-x \)[/tex] to get [tex]\( f(-x) \)[/tex], we have to consider how this affects the domain.
- For [tex]\( f(x) \)[/tex], [tex]\( x \)[/tex] ranges from [tex]\(-2\)[/tex] to [tex]\( 3 \)[/tex].
- For [tex]\( f(-x) \)[/tex], we replace [tex]\( x \)[/tex] with [tex]\(-x \)[/tex], which means we need to reverse the sign of the bounds of the domain.
- The original domain [tex]\([-2, 3]\)[/tex] will transform to [tex]\([-(3), -(-2)]\)[/tex], which simplifies to [tex]\([-3, 2]\)[/tex].
Thus, the new domain of [tex]\( f(-x) \)[/tex] is [tex]\([-3, 2]\)[/tex].
### Step 2: Determine the Range of [tex]\( f(-x) \)[/tex]
The range of [tex]\( f(x) \)[/tex] is given as [tex]\([0, 3]\)[/tex]. Changing [tex]\( x \)[/tex] to [tex]\(-x \)[/tex] does not affect the range because the output values of the function remain the same for corresponding [tex]\( x \)[/tex] values.
Therefore, the range of [tex]\( f(-x) \)[/tex] remains [tex]\([0, 3]\)[/tex].
### Conclusion
The domain of [tex]\( f(-x) \)[/tex] is [tex]\([-3, 2]\)[/tex] and the range of [tex]\( f(-x) \)[/tex] is [tex]\([0, 3]\)[/tex].
