To find the domain of
f∘g, where
f(x)=logxand
g(x)=, we need to determine the values of
xfor which
g(x)>0because the natural logarithm
f(x)=logx is only defined for positive
x.
Let's start by analyzing the expression for
g(x):
g(x)= Step 1: Determine when the denominator is not zero..
We need
2x2−2x+1≠0. The quadratic equation:
2x2−2x+1=0has a discriminant
∆=(−2)2−4×2×1=4−8=−4.
Since the discriminant is negative, the quadratic has no real roots. Thus, the denominator is never zero for any real
x.
Step 2: Find where
g(x)>0.
Since the denominator does not change sign, we only need to check when the numerator is positive:
x4−2x3+3x2−2x+2>0Notice that if we substitute
x=0:
g(0)==2For large positive and negative
x, the
x4term dominates, implying that the entire expression is positive.
Checking critical points or derivatives might be complex here due to the polynomial degrees, but observing the polynomial's end behavior suggests positivity over all reals.
Thus, the entire range of
x satisfies
g(x)>0.
**Conclusion: The domain of the composition
f∘g is
Rsince
g(x)>0 for all real values of
x.
