f(x)=-4x-2 & g(x)=5x-6. Find (f*g) and state domain.
f(x)-x^2-1 what is the equation for f^-1(x)?
f(x)=3x+2, find f(f^-1(14)).
f(x)=4x+7 and g(x)=3x-5 find (f*g)(-4)
f(x)=sqrt x+3 what is equation for f^-1(x)?
Graph y=sqrt x-2 +5 which point lies on graph?
A(7,6) B(0,5) C(-2,5) D(3,6)
The graph y=sqrt x is shifted 6 points down and 1 unit right. Which equation represents the new graph.
y=sqrt x+1 -6
y=sqrt x-6 +1
y= sqrt x-1 -6
y= sqrt x+6-1
any help would be appreciated.
I am not going to do all of these but will show you how to do one.
Your question is
f(x) = x^2-1
what is the equation for f^-1(x)?
f^-1(x) is called the inverse function of f(x). It is the function which gives you back "x" when you apply it to f(x)
Let's call f(x) just y, to make the notation look easier.
y = x^2 -1
Solve for x
x= sqrt (y+1)
Now switch the variables
y = sqrt (x+1)
That is the inverse function of f(x). That's all there is to it!
Proof: If I take sqrt(x+1), square it, and subtract 1, what do I end up with? x. Therefore applying f(x) to f^-1(x) gives me back x.
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.