Just Plug It In

Algebra Level 5

A real function $f : \mathbb{R} \to \mathbb{R}$ satisfies $f(x^2 + 3x + 4) = 2x^2 + 6x + 10$ for all real $x$ . What is the value of $f(0)$ ?

Not enough information 0 2 10 None of the others

2 solutions

Ivan Koswara
Feb 24, 2016

Since $f(x^2 + 3x + 4) = 2x^2 + 6x + 10 = 2(x^2 + 3x + 4) + 2$ , we can just plug $x^2 + 3x + 4 = 0$ and get $f(0) = 2(0) + 2 = 2$ , isn't it?

The answer is not enough information . The problem is that $x^2 + 3x + 4 = 0$ has no solution in the reals, so the information we have is not enough to determine it. It is true that if $x^2 + 3x + 4 = y$ then $f(y) = 2y+2$ , but this doesn't apply for all $y$ , only for all $y \ge \frac{7}{4}$ (the range of $x^2 + 3x + 4 = \left( x + \frac{3}{2} \right)^2 + \frac{7}{4}$ ). $f(0)$ is not defined in this way, so there's no restriction applied to it yet.

The option "None of the others" better suits here in my opinion.

We have the information that $\large x^2 + 3x + 4=0$ .

But since f(0) can't be attained for any real x, we can infer that no option given here is right.

Pulkit Gupta - 5 years, 3 months ago

Yeah that's what I picked too thinking that's what the option meant.

Vishnu Bhagyanath - 5 years, 3 months ago

The only information is for $f(x^2+3x+4)$ . There is no information for $f(0)$ . On the other hand, since the domain of $f$ includes 0, then $f(0)$ is defined. Thus it is "not enough information".

Ivan Koswara - 5 years, 3 months ago

I am not saying that your reasoning is wrong.

Its just I believe that the community may confuse itself between these two options.

Pulkit Gupta - 5 years, 3 months ago

@Pulkit Gupta – I was expecting this problem shoot to Level 5. And it did :-)

Pulkit Gupta - 5 years, 3 months ago

None of the others -> also means negation of the statement ('Not enough information'). Which is not possible because, f(y) is not defined for y<= 7/4.

Siva Bathula - 5 years, 3 months ago

This nice problem fooled me with the definition, thank you

Hjalmar Orellana Soto - 5 years, 3 months ago

Somehow my mind omitted the 4 in $b^2-4ac$ . Very smart.

Jake Lai - 5 years, 2 months ago

Vineet PaHurKar
May 16, 2016

After solving function we get the value 2 but it was wrong So,for f(0) value of is complex sondata is not enough to solve this question.... ok bro

Just Plug It In

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