i) Predict whether the functions in each pair are equivalent by testing three different values of x. (3 marks each - show your work)
ii) Determine whether the functions in each pair are equivalent by simplifying the expression on the right side. (2 marks each = show your work)
a) f(x)=-3(x+2)^2-(x+1),g(x)=-3x^2-13x-13
b)f(x)=x^2-x-2/3x^2=4x+1, g(x)=x-2/3x+1
Please help!
a. X = 2.
F(x) = -3(2+-2)^2 - (2+1) = -51
g(x) = -3*2^2 - 13*2 - 13 = -51
The student should finish the assignment.
Sure! I'd be happy to help you with these questions.
i) To predict whether the functions in each pair are equivalent, we need to substitute three different values of x into each function and compare the results.
a) Let's evaluate the functions f(x) and g(x) for three different values of x. To make it easier, let's choose x = -2, 0, and 1.
For f(x):
f(-2) = -3(-2+2)^2 - (-2+1) = -3(0)^2 - (-1) = -3(0) + 1 = 1
f(0) = -3(0+2)^2 - (0+1) = -3(2)^2 - 1 = -3(4) - 1 = -12 - 1 = -13
f(1) = -3(1+2)^2 - (1+1) = -3(3)^2 - 2 = -3(9) - 2 = -27 - 2 = -29
For g(x):
g(-2) = -3(-2)^2 - 13(-2) - 13 = -3(4) + 26 - 13 = -12 + 26 - 13 = 1
g(0) = -3(0)^2 - 13(0) - 13 = 0 - 0 - 13 = -13
g(1) = -3(1)^2 - 13(1) - 13 = -3(1) - 13 - 13 = -3 - 13 - 13 = -29
Comparing the results for f(x) and g(x), we can see that they are not equivalent. For example, f(-2) = 1 while g(-2) = 1, f(0) = -13 while g(0) = -13, and f(1) = -29 while g(1) = -29. Since the outputs are different for all the values tested, we can conclude that f(x) and g(x) are not equivalent.
b) Let's evaluate the functions f(x) and g(x) by simplifying the expressions on the right side.
For f(x):
f(x) = x^2 - x - 2 / 3x^2 = 4x + 1
To simplify the expression on the right side, we can start by multiplying both sides by 3x^2 to get rid of the fraction:
3x^2(f(x)) = (x^2 - x - 2)(3x^2) = (4x + 1)(3x^2)
Expanding both sides:
3x^2f(x) = 3x^4 - 3x^3 - 6x^2 = 12x^3 + 3x^2
From this, we can deduce that f(x) = 4x^3 + x^2 / 3 - 2.
Now let's compare this with g(x):
g(x) = x - (2/3x) + 1
Comparing the expressions for f(x) and g(x), we can see that they are not equivalent. They have different terms and coefficients.
ii) Determine whether the functions in each pair are equivalent by simplifying the expression on the right side:
a) We already determined that f(x) = -3(x + 2)^2 - (x + 1) and g(x) = -3x^2 - 13x - 13. There is no expression on the right side to simplify, so we need to compare the two functions as they are.
Comparing f(x) and g(x), we can see that they are not equivalent since they have different terms, coefficients, and overall structures.
b) We simplified f(x) to f(x) = 4x^3 + x^2 / 3 - 2. When comparing this with g(x) = x - (2/3x) + 1, we can see that they are not equivalent since they have different terms, coefficients, and overall structures.
In both cases, we can conclude that the functions in each pair are not equivalent based on their expressions and simplification.