A-2 Rational Function

Algebra Level 2

If f ( x 3 5 x ) = 3 x 2 x 2 f\left( \dfrac{x-3}{5-x} \right) = \dfrac{3x-2}{x-2} , find f ( 1 ) f(1) .


The answer is 5.

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2 solutions

Brian Dela Torre
Feb 3, 2016

In f( x 3 5 x \frac{x - 3}{5 - x} ), we want to make this function into f(1)

Thus, solving the problem we have to equate x 3 5 x \frac{x - 3}{5 - x} = 1, and we get x = 4

Then substituting thus value of x in the expression 3 x 2 x + 1 \frac{3x - 2}{x+1} , we have

3 x 2 x 2 \frac{3x - 2}{x - 2} = 3 ( 4 ) 2 ( 4 ) 2 \frac{3(4) - 2}{(4) - 2} = 5

Hence the answer is 5 \boxed{5}

Check your arithmetic men.

John Michael Gogola - 5 years, 4 months ago

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I believe it's right

Department 8 - 5 years, 4 months ago

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Thanks Sir Lakshya Sinha for your response.

Brian Dela Torre - 5 years, 4 months ago

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@Brian Dela Torre sir you are elder than me and please don't call me sir, it's embarrassing :-). BTW I got the problem wrong.

Department 8 - 5 years, 4 months ago
Kay Xspre
Feb 4, 2016

I prefer an alternative means. We rewrote the original equation to f ( 1 + 2 5 x ) = 3 + 4 x 2 f(-1+\frac{2}{5-x}) = 3+\frac{4}{x-2} Let y = 1 + 2 5 x y = -1+\frac{2}{5-x} . Rewrote it to x = 5 y + 1 2 x = 5-\frac{y+1}{2} , then plug in y y to the LHS and x x to the RHS. This will give f ( y ) = 3 + 8 5 y f ( x ) = 3 + 8 5 x = 23 3 x 5 x f(y) = 3+\frac{8}{5-y}\Rightarrow f(x)=3+\frac{8}{5-x} = \frac{23-3x}{5-x} Here, just plug in x = 1 x = 1 and the answer will be f ( 1 ) = 3 + 8 5 1 = 3 + 2 = 5 f(1) = 3+\frac{8}{5-1} = 3+2 = 5

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