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1.) Let f(x)=x2+1 and g(x)=x7. What is...
a.) (fg)(x)?
b.) (gf)(x)?
c.) (ff)(x)?
d.) (gg)(x)?
Solution:
a.) Compute (fg)(x)=f(g(x))=f(x7)=(x7)2+1=x7+1=x6. b.) Compute (gf)(x)=g(f(x))=g(x2+1)=(x2+1)7=x26. c.) Compute (ff)(x)=f(x2+1)=(x2+1)2+1=(x4+2x2+1)+1=x4+2x2+2. d.) Compute (gg)(x)=g(g(x))=g(x7)=x77.

2.) Decompose the function h(x)=1x25 by finding functions f(x) and g(x) such that h(x)=(fg)(x). Solution: There are many ways to do this. One way is to let f(x)=1x and g(x)=x7 which yields (fg)(x)=f(g(x))=f(x7)=1x7. Another way would be to let f(x)=1x and let g(x)=x7 so that (fg)(x)=f(g(x))=f(x7)=1x7.

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.