The Hidden Function

Algebra Level 3

{ g ( x ) = x 2 − 11 x + 30 g ( f ( x ) ) = x 4 − 14 x 3 + 62 x 2 − 91 x + 42 \begin{cases} g(x) = x^2 - 11x + 30 \\ g(f(x)) = x^4 - 14x^3 + 62x^2 - 91x + 42 \end{cases}

Let g g and f f be the monic polynomial functions that satisfy the system above. What is the value of f ( 7 ) f(7) ?


The answer is 12.

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

From the equations, g ( f ( 7 ) ) = 7 4 − 14 × 7 3 + 62 × 7 2 − 91 × 7 + 42 = 42 g(f(7)) = 7^4 - 14\times 7^3 + 62\times 7^2 - 91\times 7 + 42 = 42 .

Now let f ( 7 ) = a f(7) = a , for some real number a a .

Then g ( a ) = a 2 − 11 a + 30 = 42 g(a) = a^2 - 11a + 30 = 42 .

Thus, a 2 − 11 a − 12 = 0 a^2 - 11a -12 = 0 ; ( a − 12 ) ( a + 1 ) = 0 (a - 12)(a + 1) = 0 .

That is, f ( 7 ) = 12 f(7) = 12 or f ( 7 ) = − 1 f(7) = -1 .

However, consider f ( x ) = y f(x) = y . Then g ( f ( x ) ) = y 2 − 11 y + 30 = x 4 − 14 x 3 + 62 x 2 − 91 x + 42 g(f(x)) = y^2 -11y +30 = x^4 - 14x^3 + 62x^2 - 91x + 42

( y − ( 11 2 ) ) 2 − 0.25 = x 4 − 14 x 3 + 62 x 2 − 91 x + 42 (y - (\dfrac{11}{2}))^2 - 0.25 = x^4 - 14x^3 + 62x^2 - 91x + 42

y = ( 11 2 ) ± x 4 − 14 x 3 + 62 x 2 − 91 x + 42.25 y = (\dfrac{11}{2}) \pm \sqrt{x^4 - 14x^3 + 62x^2 - 91x + 42.25}

Since f ( x ) f(x) is a monic polynomial, so the minus square root is not applicable as it will result in a leading coefficient of − 1 -1 . Likewise, for the same reason, f ( 7 ) = 5.5 + 42.25 = 5.5 + 6.5 = 12 f(7) = 5.5 + \sqrt{42.25} = 5.5 + 6.5 = \boxed{12} .

Note: f ( x ) = ( 11 2 ) + x 4 − 14 x 3 + 62 x 2 − 91 x + 42.25 = 5.5 + ( x 2 − 7 x + 6.5 ) = x 2 − 7 x + 12 f(x) = (\dfrac{11}{2}) + \sqrt{x^4 - 14x^3 + 62x^2 - 91x + 42.25} = 5.5 + (x^2 - 7x + 6.5) = x^2 -7x +12 .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Your last line is wrong, x 4 − 14 x 3 + 62 x 2 − 91 x + 42.25 = ∣ x 2 − 7 x + 6.5 ∣ \sqrt{x^4 - 14x^3 + 62x^2 - 91x + 42.25} = | x^2 - 7x+6.5 | .

A simpler standard way to solve this is to find the values of a a and b b satisfying the identity,

( x 2 + a x + b ) 2 − 11 ( x 2 + a x + b ) + 30 = x 4 − 14 x 3 + 62 x 2 − 91 x + 42 . (x^2+ax+b)^2 - 11(x^2 + ax+b) + 30 = x^4-14x^3+62x^2-91x+42 \; .

Pi Han Goh - 5 years, 1 month ago

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