Find the sum and difference functions f + g and f – g for the functions given.
f(x) = 2x + 6 and g(x) = 2x - 6
f(x) = x^2 - x and g(x) = -3x + 1
f(x) = 3x^3 - 4 and g(x) = -x^2 + 3
State the domain and range for the sum and difference functions in #3.
Find the product and the quotient functions for the functions given:
f(x) = 2x + 6 and g(x) = x + 3
f(x) = x^2 + 3x - 4 and g(x) = x - 1
f(x) = sqrt(x+1) and g(x)= sqrt(x-1)
State the domain and range for each resulting function in #5.
Please explain and show all work. Math isn't my best topic!!!
they give you f and g. Is there some difficulty finding f+g, etc.?
Post what you got, and where you get stuck.
Sure! I'll help you with all the questions step by step.
Question 1:
To find the sum and difference functions, we can simply add or subtract the two functions f(x) and g(x) together.
a) For f(x) = 2x + 6 and g(x) = 2x - 6:
Sum function (f + g): (2x + 6) + (2x - 6) = 4x
Difference function (f - g): (2x + 6) - (2x - 6) = 12
b) For f(x) = x^2 - x and g(x) = -3x + 1:
Sum function (f + g): (x^2 - x) + (-3x + 1) = x^2 -4x + 1
Difference function (f - g): (x^2 - x) - (-3x + 1) = x^2 + 2x - 1
c) For f(x) = 3x^3 - 4 and g(x) = -x^2 + 3:
Sum function (f + g): (3x^3 - 4) + (-x^2 + 3) = 3x^3 - x^2 - 1
Difference function (f - g): (3x^3 - 4) - (-x^2 + 3) = 3x^3 + x^2 - 7
Question 2:
To find the domain and range for the sum and difference functions in question 3, we need to determine the possible input values (domain) and the resulting output values (range).
a) Domain: The domain of the sum and difference functions is the same as the original functions. In this case, both f(x) and g(x) have no restrictions on the input values. So, the domain for the sum and difference functions is all real numbers.
b) Range: The range of the sum and difference functions can be determined by analyzing the resulting output values. In this case, since the sum and difference functions involve only algebraic manipulation, the range would be all real numbers as well.
Question 3:
To find the product and quotient functions, we need to multiply and divide the two given functions f(x) and g(x).
a) For f(x) = 2x + 6 and g(x) = x + 3:
Product function (f * g): (2x + 6) * (x + 3) = 2x^2 + 12x + 18
Quotient function (f / g): (2x + 6) / (x + 3) = 2
b) For f(x) = x^2 + 3x - 4 and g(x) = x - 1:
Product function (f * g): (x^2 + 3x - 4) * (x - 1) = x^3 + 2x^2 - 7x + 4
Quotient function (f / g): (x^2 + 3x - 4) / (x - 1) = x + 4
c) For f(x) = sqrt(x+1) and g(x) = sqrt(x-1):
Product function (f * g): (sqrt(x+1)) * (sqrt(x-1)) = sqrt((x+1)(x-1))
Quotient function (f / g): (sqrt(x+1)) / (sqrt(x-1))
Question 4:
To determine the domain and range for each resulting function, we need to consider any restrictions on the input values and the resulting output values.
a) For the product function in 5a:
Domain: The domain is restricted by the square root function, which requires the expression inside the square root (x+1)(x-1) to be non-negative. Therefore, x + 1 ≥ 0 and x - 1 ≥ 0. Solving these inequalities, we find x ≥ -1 and x ≥ 1, so the domain is x ≥ 1.
Range: The range of the product function is all non-negative real numbers.
b) For the quotient function in 5b:
Domain: The domain is restricted by the square root function, which requires the expression inside the square root (x-1) to be non-negative. Therefore, x - 1 ≥ 0, which leads to x ≥ 1. So, the domain is x ≥ 1.
Range: The range of the quotient function is all real numbers.
c) For the quotient function in 5c:
Domain: The domain is restricted by the square root function, which requires the expression inside the square root (x-1) to be non-negative. Therefore, x - 1 ≥ 0, which leads to x ≥ 1. So, the domain is x ≥ 1.
Range: The range of the quotient function is all real numbers.
I hope this explanation helps! Let me know if you have any further questions.