Operations of Functions
f(x) = (x^2 + 2x - 8) and g(x) = (6x^2 -10x + 3), find (f+g)(x)
Answer: 7x^2 - 8x - 5
Work :
x^2+2x-8+6x^2-10x+3
Simplify:x^2+2x-8+6x^2-10x+3:
7x^2-8x-5
Find the inverse : g(x)= − x
Answer: f-1(x) = -x
Replace x and y
x = -y
Divide both sides by -1
y= -x
Find the Domain of f(x) = (x - 1)/(x - 3)
Answer: the domain is a set of all real numbers except 3
Work:
To get the values for which the function is undefined, we must equate the denominator to 0.
x - 3 = 0
x = 3
Clearly, for the value x = 3 the function will become undefined. So, the domain is set of all real numbers except 3.
Solve the following function: 11=x^2−5
Answer: x=4 or x=−4.
Work:
You must get x by itself so you must add 5 to both side which results in
16=x^2.
You must get the square root of both side to undue the exponent.
This leaves you with x=4.
But since you square the x in the equation, the original value you plug can also be its negative value since squaring it will make it positive anyway.
This means your answer can be x=4 or x=−4.
Solve this quadratic equation - u^2−5u−14=0
Answer: u=−2 and u=7
Work:
u+2=0
u=−2
OR
u−7=0
u=7
f(x) = 2x^2 + 2x and g(x) = x + 1, find (fg)(x)
Answer: 2x^3+4x^2+2x
Work:
(2x^2+2x)(x+1)
Simplify using GDC: 2x^3+4x^2+2x
Find the Inverse of f(x) = (x+1)/(x-1)
Answer : f-1(x) = (1+x) / (x-1)
Work:
Replace x and y
x = (1+y) / (y-1)
Use a GDC = f-1(x) = (1+x) / (x-1)
Find the Domain of f(x) = (2x - 3)/(x^2 - 3x + 2)
Answer: R - {1, 2} is the domain of the given function.
Work:
In order to find domain, let us equate the denominator equal to 0.
x^2 - 3x + 2 = 0
(x - 1) (x - 2) = 0
x = 1 and x = 2
Hence R - {1, 2} is the domain of the given function.
Solve 101−x=104
Answer: x = −3
Work:
Since the bases are the same, then I can equate the powers and solve:
1 − x = 4
1 − 4 = x
−3 = x
The solution is:
x = −3
Solve this quadratic equation - y^2=11y−28
Answer: y=4 and y=7
Work:
y^2−11y+28=0
(y−4)(y−7)=0
y−4=0
y=4
ORy−7=0
y=7
f(x) = x^2 - 7x + 2 and g(x) = 5x^2 - 8x - 9, find (f-g)(x)
Answer: -4x^2 + x + 11
Work:
x^2-7x+2-(5x^2-8x-9)
Simplify: -4x^2 + x + 11
Find the inverse of f(x) = x / (x+2
Answer: f-1(x) = -(2x) / (x-1)
Work:
Replace x with y
x = y / (y+2)
Simplify:
-(2x) / (x-1)
Find the Domain of f(x) = √(x - 2)
Answer: Domain is [2, ∞)
Work:
Domain for the radical function means, the values we choose for x must satisfy the condition f(x) ≥ 0
x - 2 ≥ 0
x ≥ 2
Solve 3(x^2−3)=81
Answer:x=−1,4
Work:
3x^2−3x=81
3^2−3=34
x^2−3x=4
x^2−3x−4=0
(x−4)(x+1)=0
x=−1,4
Solve this quadratic equation - 19x=7−6x2
Answer - x=1/3 and x=−7/2
Work-
6x2+19x−7=0
(3x−1)(2x+7)=0
3x−1=0
x=1/3
OR
2x+7=0
x=−7/2
f(x) = x^2 + 5x + 6 and g(x) = 2x + 4, find (f/g)(x).
Answer: x+3/2
Work: x^2+5x+6/2x+4
facotr x^2+5x+6: (x+2)(x+3)
(x+2)(x+3)/2x+4
factor 2x+4: (2x+2)
Remove the common factor of x+2
simplify:
x+3/2
Find the inverse of f(x) = (10-x) / 5
Answer: f-1(x) = -5x+10
Work:
Replace x with y : x= (10-y) / 5
Time 5 to both sides, and then subtract 10 on both sides
-5x+10
Find the Domain of f(x) = (2x + 1)/(x^2 - 9)
Answer: R - [-3, 3]
Work:
f(x) = (2x + 1)/(x^2 - 9)
f(x) = (2x + 1)/(x + 3)(x - 3)
(x + 3)(x - 3) = 0
x = -3 and x = 3
Solve 3^(2−1)=27
Answer: x = 2
Work:
Since 27 = 3^3, then proceed with the solution:
3^2x−1 = 27
3^2x−1 = 33
2x − 1 = 3
2x = 4
x=2
Solve this quadratic equation - z^2−16z+61=2z−20
Answer: z=9
Work:
z2−18z+81=0
(z−9)2=0
z=9