Question

# Polynomial calculations Find a polynomial f that satisfies the following properties. Determine the degree of f. then substitute a polynomial of that degree and solve for its coefficients. f(f(x))=9x-8 f(f(x))=x^{4}-12x^{2}+30

Applications of integrals
Polynomial calculations Find a polynomial f that satisfies the following properties. Determine the degree of f. then substitute a polynomial of that degree and solve for its coefficients.
$$\displaystyle{f{{\left({f{{\left({x}\right)}}}\right)}}}={9}{x}-{8}$$
$$\displaystyle{f{{\left({f{{\left({x}\right)}}}\right)}}}={x}^{{{4}}}-{12}{x}^{{{2}}}+{30}$$

2021-04-23

Step 1
1) Given that,
$$f(f(x))=9x-8...$$(1)
Step 2
Since degree of given composition is 1 so degree of f is also 1.
Consider,
$$f(x)=ax+b$$
So
$$(f \cdot f)(x)=f(f(x))$$
$$=f(ax+b)$$
$$=a(ax+b)+b (Since\ f(x)=ax+b)$$
$$\displaystyle={a}^{{{2}}}{x}+{a}{b}+{b}$$...(2)
Step 3
Using (1) and (2),
$$\displaystyle{9}{x}-{8}={a}^{{{2}}}{x}+{a}{b}+{b}$$
Compare the coefficient of x and constant term,
$$\displaystyle{a}^{{{2}}}={9}\Rightarrow{a}=\pm{3}$$
And
$$\displaystyle{a}{b}+{b}=-{8}\Rightarrow{b}=-{\frac{{{8}}}{{{a}+{1}}}}$$
When $$a=3, b=-2$$
When $$a=-3, b=4$$
So
When $$a=3$$ and $$b=-2\ then\ f(x)=3x-2$$
When $$a=-3$$ and $$b=4\ then\ f(x)=-3x+4$$
Step 4
Hence,
$$f(x)=3x-2$$
$$f(x)=-3x+4$$