1. find the solution
sqrt 2x - 8 + 2 = 4
2. let f(x) = 3x+6 and g(x) and = -x-4. find f(x) - g(x)
3. find the inverse of the function.
4. rewrite the expression to make it easier to graph. what are the transformations to the parent function?
^3 sqrt 8x - 24 + 4
5. suppose f(x) = 3x-1 and g(x) = 4x+3. find f(g(2)).
1. To find the solution, we will isolate the variable x:
√(2x - 8) + 2 = 4
First, we'll subtract 2 from both sides:
√(2x - 8) = 2
Then, we'll square both sides:
2x - 8 = 4
Next, we'll add 8 to both sides:
2x = 12
Finally, we'll divide both sides by 2:
x = 6
Therefore, the solution to the equation is x = 6.
2. To find f(x) - g(x), we need to substitute the functions f(x) and g(x) into the expression:
f(x) - g(x) = (3x + 6) - (-x - 4)
Simplifying the expression:
f(x) - g(x) = 3x + 6 + x + 4
Combining like terms:
f(x) - g(x) = 4x + 10
Therefore, f(x) - g(x) = 4x + 10.
3. To find the inverse of a function, we swap the dependent and independent variables and solve for the dependent variable. Let's denote the original function as y = f(x):
y = 3x - 1
Now, let's swap x and y:
x = 3y - 1
Next, we'll solve for y:
x + 1 = 3y
Dividing both sides by 3:
y = (x + 1)/3
Therefore, the inverse of the function f(x) = 3x - 1 is f^(-1)(x) = (x + 1)/3.
4. The given expression is ∛(8x - 24) + 4. To make it easier to graph, we can simplify it further:
∛(8x - 24) + 4 = ∛8(x - 3) + 4
The parent function is ∛x. The transformation done are:
1. Shifting the graph horizontally to the right by 3 units (x - 3 in the inside).
2. Stretching the graph vertically by a factor of 8 (8(x - 3)).
3. Shifting the graph vertically upwards by 4 units (+4 at the end).
5. To find f(g(2)), we first need to find g(2) and then substitute it into f(x).
g(x) = 4x + 3, so g(2) = 4(2) + 3 = 11.
Now, substitute g(2) into f(x):
f(g(2)) = f(11)
f(x) = 3x - 1, so f(11) = 3(11) - 1 = 33 - 1 = 32.
Therefore, f(g(2)) = 32.