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1.) Let f(x)=x2+1 and g(x)=√x−7. What is...
a.) (f∘g)(x)?
b.) (g∘f)(x)?
c.) (f∘f)(x)?
d.) (g∘g)(x)?
Solution:
a.) Compute
(f∘g)(x)=f(g(x))=f(√x−7)=(√x−7)2+1=x−7+1=x−6.
b.) Compute
(g∘f)(x)=g(f(x))=g(x2+1)=√(x2+1)−7=√x2−6.
c.) Compute
(f∘f)(x)=f(x2+1)=(x2+1)2+1=(x4+2x2+1)+1=x4+2x2+2.
d.) Compute
(g∘g)(x)=g(g(x))=g(√x−7)=√√x−7−7.
2.) Decompose the function h(x)=1√x2−5 by finding functions f(x) and g(x) such that
h(x)=(f∘g)(x).
Solution: There are many ways to do this. One way is to let f(x)=1x and g(x)=√x−7 which yields
(f∘g)(x)=f(g(x))=f(√x−7)=1√x−7.
Another way would be to let f(x)=1√x and let g(x)=x−7 so that
(f∘g)(x)=f(g(x))=f(x−7)=1√x−7.
3.) Do the following graphs exhibit any symmetry? If so, what kind(s)?
a.)
b.)
c.)
Solution:
a.) Symmetric with respect to the x-axis.
b.) Symmetric with respect to the x-axis, with respect to the y-axis, and with respect to the origin.
c.) Symmetric with respect to the y-axis.