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

so, f is defined for x > -4 and y < -1.

2*f(x) has same domain, range is y < -2.
f(-x) has domain (-x) > -4, so x < 4, range y < -1.

3*f(x+1) + 4 has domain (x+1) > -4, so x > -5, range y < 3*R(f)+4 so y < 1.

-2*f(-x+5)+1 has domain (-x+5) > -4, so x < 9. range would be -2*R+1 = y>3

yo what

I'm so confused. can u perhaps explain a bit better since I'm pretty dumb

We all on page 77 question 19 aren't we TT

a) For function y = 2f(x), the domain remains the same as the original function f(x), which is x > -4. The range is multiplied by 2, so it becomes y < -2.

b) For function y = f(-x), the domain is found by replacing x with -x in the original domain, so it becomes x < 4. The range remains the same as the original function f(x), which is y < -1.

c) For function y = 3f(x+1) + 4, the domain remains the same as the original function f(x), which is x > -4. The range is shifted upwards by 4 units, so it becomes y < 3.

d) For function y = -2f(-x+5) + 1, the domain is found by replacing x with -x+5 in the original domain, so it becomes x < -1. The range is multiplied by -2 and shifted upwards by 1 unit, so it becomes y > 3.

To determine the domain and range of each function, we need to consider the given domain and range of the original function, f(x), and how it is affected by the given transformations.

a) y = 2f(x):
- Domain: The domain of y = 2f(x) is the same as the domain of f(x), which is { x ∈ ℝ | x > -4 }.
- Range: To determine the range of y = 2f(x), we need to consider the range of f(x) and how it is scaled by the factor of 2. Since the range of f(x) is { y ∈ ℝ | y < -1 }, multiplying all the values in the range by 2 will result in the range of y becoming { y ∈ ℝ | y < -2 }.

Therefore, the domain of y = 2f(x) is { x ∈ ℝ | x > -4 } and the range is { y ∈ ℝ | y < -2 }.

b) y = f(-x):
- Domain: The domain of y = f(-x) is obtained by reflecting the original domain of f(x) across the y-axis. Thus, the domain becomes { x ∈ ℝ | x < 4 }.
- Range: The range of y = f(-x) remains the same as the range of f(x), which is { y ∈ ℝ | y < -1 }.

Therefore, the domain of y = f(-x) is { x ∈ ℝ | x < 4 } and the range is { y ∈ ℝ | y < -1 }.

c) y = 3f(x+1) + 4:
- Domain: The original domain of f(x) is { x ∈ ℝ | x > -4 }. Adding 1 to x in the function y = 3f(x+1) shifts the graph of f(x) one unit to the left. Therefore, the domain becomes { x ∈ ℝ | x > -5 }.
- Range: The original range of f(x) is { y ∈ ℝ | y < -1 }. Multiplying all the values in the range by 3 and adding 4 shifts the graph vertically upward. Thus, the range becomes { y ∈ ℝ | y < 11 }.

Therefore, the domain of y = 3f(x+1) + 4 is { x ∈ ℝ | x > -5 } and the range is { y ∈ ℝ | y < 11 }.

d) y = -2f(-x+5) + 1:
- Domain: The original domain of f(x) is { x ∈ ℝ | x > -4 }. Shifting the graph of f(x) by adding 5 to x in the function y = -2f(-x+5) moves the graph 5 units to the right. Thus, the domain becomes { x ∈ ℝ | x > 1 }.
- Range: The original range of f(x) is { y ∈ ℝ | y < -1 }. Multiplying all the values in the range by -2, adding 1, and reflecting the graph vertically across the x-axis results in the range becoming { y ∈ ℝ | y > 1 }.

Therefore, the domain of y = -2f(-x+5) + 1 is { x ∈ ℝ | x > 1 } and the range is { y ∈ ℝ | y > 1 }.

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 > -5 } Range: { y ∈ ℝ | y < 11 }
d) Domain: { x ∈ ℝ | x > 1 } Range: { y ∈ ℝ | y > 1 }