Suppose f(x)= 3x+4, g(x)=2x^2+3x, & h(x)=-x+6. Find each of the following and write all polynomials in standard form.

Question

a. f o g o h(2)

b. f o f o f(x)

c. f o g o h(x)

d. f^-1(x)

e. h^-1(x)

f. f^-1 o f(x)

g. f(h(h^-1(f^-1(x))))

b.f◦f

Answer 1

a. fºgºh(2)=fºg(-2+6)

=fºg(4)

=f(2⋅4²+3⋅4)

=f(2⋅16+3⋅4)

=f(32+12)

=f(44)

=3⋅44+4

=132+4

=136

b. fºfºf(x)=fºf(3x+4)

=f[3(3x+4)+4]

=f(9x+12+4)

=f(9x+16)

=3(9x+16)+4

=27x+48+4

=27x+52

c. fºgºh(x)=fºg(-x+6)

=f[2(-x+6)²+3(-x+6)]

=f[2(x²-12x+36)+3(-x+6)]

=f(2x²-24x+72-3x+18)

=f(2x²-27x+90)

=3(2x²-27x+90)+4

=6x²-81x+270+4

=6x²-81x+274

d. 3f^-1(x)+4=x

3f^-1(x)+4-4=x-4

3f^-1(x)=x-4

[3f^-1(x)]/3=(x-4)/3

f^-1(x)=(1/3)x-4/3

e. -h^-1(x)+6=x

-h^-1(x)+6-6=x-6

-h^-1(x)=x-6

[-h^-1(x)]/(-1)=(x-6)/(-1)

h^-1(x)=-x+6

f. f^-1ºf(x)=x by definition, but in case you need to evaluate this, here goes:

f^-1ºf(x)=f^-1(3x+4)

=(1/3)(3x+4)-4/3

=x+4/3-4/3

=x

g. f[h(h^-1[f^-1(x)])]=f[h(h^-1[(1/3)x-4/3])]

=f[h(-[(1/3)x-4/3]+6)]

=f(h[(-1/3)x+4/3+6])

=f(h[(-1/3)x+22/3])

=f(-[(-1/3)x+22/3]+6)

=f[(1/3)x-22/3+6]

=f[(1/3)x-4/3]

=3[(1/3)x-4/3]+4

=x-4+4

=x