To find (f * g) and determine its domain, we need to multiply the two functions f(x) and g(x):
(f * g)(x) = f(x) * g(x)
Plugging in the given functions:
(f * g)(x) = (-4x - 2) * (5x - 6)
= -20x^2 + 24x + 8
The domain of (f * g)(x) is the set of all real numbers since there are no restrictions on the variables.
To find the equation for f^-1(x), we need to swap the x and y variables and then solve for y:
f(x) = -x^2 - 1
Swap x and y:
x = -y^2 - 1
Solving for y:
y^2 = -x - 1
Taking the square root of both sides, we get:
y = ±√(-x - 1)
The equation for f^-1(x) is y = √(-x - 1) or y = -√(-x - 1).
To find f(f^-1(14)), we first need to find f^-1(14) and then substitute it into f(x):
f(x) = 3x + 2
Let's find f^-1(x):
x = 3y + 2
3y = x - 2
y = (x - 2) / 3
Substituting f^-1(x) = (x - 2) / 3 into f(x):
f(f^-1(14)) = f((14 - 2) / 3)
= f(12 / 3)
= f(4)
= 3(4) + 2
= 14
Therefore, f(f^-1(14)) = 14.
To find (f * g)(-4), we substitute -4 into the (f * g)(x) equation:
(f * g)(x) = (4x + 7) * (3x - 5)
= 12x^2 - 8x + 21x - 35
= 12x^2 + 13x - 35
Hence, (f * g)(-4) = 12(-4)^2 + 13(-4) - 35 = 192 - 52 - 35 = 105.
To find the equation for f^-1(x) for f(x) = √x + 3, we swap x and y then solve for y:
f(x) = √x + 3
Swap x and y:
x = √y + 3
Solving for y:
√y = x - 3
Square both sides:
y = (x - 3)^2
Therefore, the equation for f^-1(x) is y = (x - 3)^2.
To determine which point lies on the graph y = √x - 2 + 5, we need to substitute the values of x and y for each given point and check if the equation holds true.
For point A(7,6):
6 = √7 - 2 + 5
This equation is not true since √7 - 2 is less than 1.
For point B(0,5):
5 = √0 - 2 + 5
This equation is true.
For point C(-2,5):
5 = √(-2) - 2 + 5
This equation is not true since √(-2) is not a real number.
For point D(3,6):
6 = √3 - 2 + 5
This equation is true.
Therefore, the point that lies on the graph of y = √x - 2 + 5 is B(0,5).
To determine the equation representing the new graph after shifting the graph y = √x, we need to adjust the square root function accordingly.
The original graph y = √x is shifted 6 units down and 1 unit right. Thus, we need to replace x with (x - 1) and subtract 6 from the entire equation:
y = √(x - 1) - 6
Therefore, the equation representing the new graph is y = √(x - 1) - 6.
I hope this explanation helps you understand how to find the answers to these questions! If you have any further questions, feel free to ask.