Determine if the function f(x) is the simplified form of g(x). If it is, state the non-permissible values. If it is not, determine the corrected simplified form, including the non-permissible values. (2 marks - show your work)
g(x)=(6x^2)+(x-12)/(2x+3),f(x)=3x-4
To determine if f(x) is the simplified form of g(x), we can compare the two functions and check if they are equivalent.
g(x) = (6x^2 + x - 12) / (2x + 3)
f(x) = 3x - 4
The two functions are not equivalent as they have different expressions.
To find the simplified form of g(x), we need to simplify the expression (6x^2 + x - 12) / (2x + 3).
To do this, we can use polynomial long division.
3x - 5
__________
2x + 3 | 6x^2 + x - 12
- (6x^2 + 9x)
___________
-8x - 12
+ (8x + 12)
___________
0
After performing the polynomial long division, we get:
g(x) = 3x - 5
Now let's find the non-permissible values for g(x).
Non-permissible values are the values of x that make the denominator (2x + 3) equal to zero since division by zero is undefined.
To find the non-permissible values, we solve the equation 2x + 3 = 0 for x:
2x + 3 = 0
2x = -3
x = -3/2
Therefore, the non-permissible value for g(x) is x = -3/2.
In conclusion:
- The function f(x) is not the simplified form of g(x).
- The simplified form of g(x) is g(x) = 3x - 5.
- The non-permissible value for g(x) is x = -3/2.
To determine if the function f(x) is the simplified form of g(x), we need to compare the two functions and check if they are equivalent.
First, let's compare the simplified form of g(x) and f(x):
g(x) = (6x^2 + x - 12) / (2x + 3)
f(x) = 3x - 4
As we can see, g(x) contains an additional term in the numerator compared to f(x), so f(x) is not the simplified form of g(x).
To find the simplified form of g(x), we need to simplify the expression (6x^2 + x - 12) / (2x + 3).
However, before simplifying, we need to consider the non-permissible values. Non-permissible values occur when the denominator of a fraction is equal to zero because division by zero is undefined.
So, to find the non-permissible values, we set the denominator, 2x + 3, equal to zero and solve for x:
2x + 3 = 0
2x = -3
x = -3/2
Thus, the non-permissible value for g(x) is x = -3/2.
Now, let's simplify g(x):
To simplify the expression (6x^2 + x - 12) / (2x + 3), we can use polynomial long division or factor out a common factor, but in this case, polynomial long division seems more straightforward.
Performing polynomial long division, we get:
3x - 4
___________
2x + 3 | 6x^2 + x - 12
6x^2 + 9x
______________
-8x - 12
-8x - 12
__________
0
The result of the division is 3x - 4, which confirms that f(x) = 3x - 4 is the simplified form of g(x) = (6x^2 + x - 12) / (2x + 3).
However, we need to keep in mind that the non-permissible value x = -3/2 still applies to the simplified form f(x) = 3x - 4.
In summary, f(x) = 3x - 4 is the simplified form of g(x) = (6x^2 + x - 12) / (2x + 3), and the non-permissible value is x = -3/2.
if f(x) = g(x), it must be true for all values of x
(all we need is ONE exception and they would not be equal)
f(1) = 6 + -11/5 = a fraction
g(1) = 3-4 = -1
so f(x) is NOT the simplified form of g(x)