Use the given functions to find g(f(x)), and give the restrictions on x.
f(x)=1/x-4
g(x)=(3/x)+4
I'm having trouble find the restrictions for x
I assume you made a parenthesis mistake and mean
1 / (x-4)
g( f(x) )=(3/{ 1/(x -4) } )+4
g( f(x) )=(3( x-4) )+4
= 3 x - 12 + 4
= 3 x - 8
I got that part correct. Thank you!
But how do I look for the restrictions on x?
So where x=/= ?
When x = 4 you have a zero denominator
Then you have
g( f(x) )=(3/{ 1/ (ZERO) } )+4
So restrictions would be 4 and 0?
I don't know why this is confusing me so much!!!
Just 4
because x=4 gives 4-4 = 0 in the denominator
To find the restrictions on x for the function g(f(x)), we need to consider the restrictions for both f(x) and g(x) individually.
Let's start with f(x). The only restriction we need to be aware of is that the denominator cannot be zero, as division by zero is undefined. In this case, the denominator is x - 4 in the function f(x). Therefore, we need to ensure that x - 4 ≠ 0.
To find the value that makes x - 4 equal to zero, we set x - 4 = 0 and solve for x:
x - 4 = 0
x = 4
So, x = 4 is the restriction for f(x).
Next, let's consider the function g(x). Again, the only restriction we need to consider is when the denominator equals zero. In this case, the denominator is x in the function g(x). Therefore, we need to ensure that x ≠ 0.
So, the restriction for g(x) is x ≠ 0.
To find g(f(x)), we substitute f(x) into g(x):
g(f(x)) = g(1/(x - 4)).
Now we need to substitute the expression for f(x) into g(x), which gives us:
g(f(x)) = g(1/(x - 4)) = 3/(1/(x - 4)) + 4.
To simplify further, we can multiply the numerator and denominator of the fraction by (x - 4) to get rid of the fraction in the denominator:
g(f(x)) = g(1/(x - 4)) = 3*(x - 4) + 4*(x - 4).
Now we can simplify:
g(f(x)) = g(1/(x - 4)) = 3x - 12 + 4x - 16.
Combining like terms:
g(f(x)) = g(1/(x - 4)) = 7x - 28.
So, g(f(x)) = 7x - 28.
In summary, the restrictions on x for g(f(x)) are x ≠ 0 (from g(x)) and x ≠ 4 (from f(x)).