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)
So, can you not just plug in some values and check the results? I'll show you my work if you'll show me yours...
Note: Usually it's easiest to use -1,0,1 for checking stuff.
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i) To predict whether the functions in each pair are equivalent, we need to test three different values of x in each function and compare the outputs.
a) For the first pair of functions, f(x) = -3(x+2)^2 - (x+1) and g(x) = (-3x^2) - 13x - 13:
Let's test the functions with three different values of x:
For x = 0:
f(0) = -3(0+2)^2 - (0+1) = -3(2)^2 - 1 = -3(4) - 1 = -12 - 1 = -13
g(0) = (-3(0)^2) - 13(0) -13 = 0 - 0 - 13 = -13
For x = 1:
f(1) = -3(1+2)^2 - (1+1) = -3(3)^2 - 2 = -3(9) - 2 = -27 - 2 = -29
g(1) = (-3(1)^2) - 13(1) - 13 = -3 - 13 - 13 = -29
For x = -2:
f(-2) = -3(-2+2)^2 - (-2+1) = -3(0)^2 - 1 = -3(0) - 1 = -1
g(-2) = (-3(-2)^2) - 13(-2) - 13 = -3(4) + 26 - 13 = -12 + 26 - 13 = 1
From the test values above, we can see that f(x) and g(x) do not have the same outputs for all values of x. Therefore, they are not equivalent.
b) For the second pair of functions, f(x) = (x^2 - x - 2)/(3x^2 + 4x + 1) and g(x) = (x - 2)/(3x + 1):
Let's simplify the expressions on the right side of each function to check if they are equivalent:
f(x) = (x^2 - x - 2)/(3x^2 + 4x + 1)
g(x) = (x - 2)/(3x + 1)
Now, let's simplify both expressions:
f(x) = (x - 2)(x + 1)/((x + 1)(3x + 1))
= (x - 2)/(3x + 1)
g(x) = (x - 2)/(3x + 1)
From the simplification, we can see that the expressions on the right side are the same. Therefore, f(x) and g(x) are equivalent.
ii) To determine whether the functions in each pair are equivalent, we need to simplify the expression on the right side for each function and compare them.
a) For the first pair of functions, f(x) = -3(x+2)^2 - (x+1) and g(x) = (-3x^2) - 13x - 13:
Let's simplify the expressions:
f(x) = -3(x+2)^2 - (x+1)
= -3(x^2 + 4x + 4) - x - 1
= -3x^2 - 12x - 12 - x - 1
= -3x^2 - 13x - 13
g(x) = (-3x^2) - 13x - 13
By comparing the simplified expressions, we can see that f(x) and g(x) are equivalent.
b) For the second pair of functions, f(x) = (x^2 - x - 2)/(3x^2 + 4x + 1) and g(x) = (x - 2)/(3x + 1):
Let's simplify the expressions:
f(x) = (x^2 - x - 2)/(3x^2 + 4x + 1)
= (x - 2)(x + 1)/((x + 1)(3x + 1))
= (x - 2)/(3x + 1)
g(x) = (x - 2)/(3x + 1)
By comparing the simplified expressions, we can see that f(x) and g(x) are equivalent.