Find the exact value of cot S in simplest form.
√31 = QS
√15 = RS
4 = QR
First, we must find the value of angle QRS using the Law of Cosines:
4² = (√31)² + (√15)² - 2(√31)(√15)cos(QRS)
16 = 31 + 15 - 2(√31)(√15)cos(QRS)
cos(QRS) = -(√31)(√15)/8
Since cot(S) is the reciprocal of tan(S), and we know that tan(S) = sin(S)/cos(S), we need to find sin(S) and cos(S):
sin(S) = RS/QS = (√15)/(√31)
cos(S) = QR/QS = 4/(√31)
Therefore, cot(S) = cos(S)/sin(S) = (4/(√31))/(√15/(√31)) = 4/√465 = (4√465)/465.
The exact value of cot(S) in simplest form is (4√465)/465.
Given the definitions of f (x) and g(x) below, find the value of (g∘f)(−1).
f(x) = -5x -3
g(x) = 3x^2 + 2x -4
First, we need to find f(-1) by substituting -1 in place of x in the equation for f(x):
f(-1) = -5(-1) -3 = 5 -3 = 2
Next, we need to find g(f(-1)) by substituting the value we just found (2) in place of x in the equation for g(x):
g(f(-1)) = g(2) = 3(2)^2 + 2(2) -4 = 12 + 4 -4 = 12
Therefore, the value of (g∘f)(−1) is 12.
Given the definitions of f (x) and g(x) find the value of g(f(1)).
f(x)= −2x+4
g(x) = 3x^2 - 7x +9
First, we need to find f(1) by substituting 1 in place of x in the equation for f(x):
f(1) = -2(1) + 4 = 2
Next, we need to find g(f(1)) by substituting the value we just found (2) in place of x in the equation for g(x):
g(f(1)) = g(2) = 3(2)^2 - 7(2) + 9 = 12 - 14 + 9 = 7
Therefore, the value of g(f(1)) is 7.
For the function f(x)= (4x)^7 find f^-1 (x)
To find the inverse of a function, we switch the roles of x and y and solve for y.
Let us begin by replacing f(x) with y:
y = (4x)^7
Then, we exchange the positions of x and y:
x = (4y)^7
Next, we isolate y:
x = (4y)^7
x^(1/7) = 4y
y = (x^(1/7))/4
Thus, we have found an expression for the inverse of f(x):
f^-1(x) = (x^(1/7))/4