Composite function: If f(x)=3x+1 and g(x)=\frac{x}{x^2+25}, how to solve g(f(x))

Danitat6

Danitat6

Answered question

2022-01-22

Composite function: If f(x)=3x+1 and g(x)=xx2+25, how to solve g(f(x))

Answer & Explanation

sphwngzt

sphwngzt

Beginner2022-01-23Added 11 answers

The first thing to do here is figure out how the expression for
(gf)(x)=g(f(x))
To do that, use f(x)=3x+1 as value for x in g(x)=xx2+25. You will get
g(f(x))=3x+1(3x+1)2+25
g(f(x))=3x+1(9x2+6x+1)+25
g(f(x))=3x+19x2+6x+26
Now, you know that you must solve
g(f(x))
which means that you have
3x+19x2+6x+26=0
As you know, a fraction can be equal to zero only if its numerator is equal to zero. In your case, this implies
3x+1=0x=13
One last thing to do here -- make sure that x=13 does not make the denominator equal to zero.
9(13)2+6(13)+26=9192+26
=12+260
Therefore, you can say that g(f(x))=0 for x=13.
As an alternative to plugging in x=13 to check if the denominator is equal to zero, notice that this quadratic equation
9x2+6x+26=0
will never actually be equal to zero because its discriminant
Δ=b24ac
Δ=624926
is negative, Δ<0. This implies that this quadratic equation doesnt

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