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
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asked 2021-04-21
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.
\(\displaystyle{f{{\left({f{{\left({x}\right)}}}\right)}}}={9}{x}-{8}\)
\(\displaystyle{f{{\left({f{{\left({x}\right)}}}\right)}}}={x}^{{{4}}}-{12}{x}^{{{2}}}+{30}\)

Answers (1)

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,
\(f(x)=3x-2\)
\(f(x)=-3x+4\)

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