a function f(x)has doamin {xeR/x>-4} and range {yeR/y<-1) . Determine the domain and range for each function.

a) y=2f(x) b) y=f(-x) c) y=3f(x+1)+4

d) y=-2f(-x+5)+1

thanks for the help !!!

To determine the domain and range for each function, we need to manipulate the given domain and range of f(x) based on the given functions.

a) y = 2f(x):
The domain remains the same as f(x), which is {x ∈ R / x > -4}.
The range is determined by multiplying the range of f(x) by 2. Since the given range is {y ∈ R / y < -1}, the new range becomes {y ∈ R / y < -2}.

b) y = f(-x):
The domain for this function is determined by applying the negation to the domain of f(x). So, x > -4 becomes -x > -4, which simplifies to x < 4. Therefore, the domain for this function is {x ∈ R / x < 4}.
The range remains the same as the range of f(x), which is {y ∈ R / y < -1}.

c) y = 3f(x + 1) + 4:
The domain remains the same as f(x), which is {x ∈ R / x > -4}.
To determine the range, we need to apply the transformation on the range of f(x). First, shift the range of f(x) one unit up (adding 1), then multiply it by 3, and finally add 4. Since the given range is {y ∈ R / y < -1}, the new range becomes {y ∈ R / y < 2}.

d) y = -2f(-x + 5) + 1:
The domain for this function is determined by applying the transformations to the domain of f(x). First, apply the negation and then shift it five units to the right. So, x > -4 becomes -x + 5 > -4, which simplifies to -x > -9 and x < 9. Therefore, the domain for this function is {x ∈ R / x < 9}.
To determine the range, we apply the transformations to the range of f(x). First, apply the negation, then shift five units to the right, multiply by -2, and finally add 1. Since the given range is {y ∈ R / y < -1}, the new range becomes {y ∈ R / y > 3}.

In summary:
a) Domain: {x ∈ R / x > -4}
Range: {y ∈ R / y < -2}

b) Domain: {x ∈ R / x < 4}
Range: {y ∈ R / y < -1}

c) Domain: {x ∈ R / x > -4}
Range: {y ∈ R / y < 2}

d) Domain: {x ∈ R / x < 9}
Range: {y ∈ R / y > 3}