What Is the Value of f(2) if f(x) = a^x and f(3) = 64?

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Discussion Overview

The discussion revolves around determining the value of f(2) given the function f(x) = a^x and the condition that f(3) = 64. Participants explore various approaches to solve for f(2), including evaluating the constant a and using properties of exponents.

Discussion Character

  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • Some participants suggest that to find f(2), one must first determine the value of a from the equation f(3) = 64.
  • One participant proposes that since 64 is a cube number, a could be inferred as a whole number, specifically 4, leading to the conclusion that f(2) = 16.
  • Another participant confirms the calculation by stating that if a = 4, then f(2) = 4^2 = 16.
  • An alternative approach is mentioned where a^2 is expressed as (a^3)^(2/3) = 64^(2/3), suggesting a different method to arrive at the value of f(2).

Areas of Agreement / Disagreement

Participants generally agree on the method to find f(2) once a is determined, but there are multiple approaches presented, and no consensus on the preferred method is established.

Contextual Notes

Some assumptions about the properties of exponents and the nature of the constant a are made, but these are not explicitly stated or resolved in the discussion.

goosey00
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If you have f(x)=a^x and f(3)=64 what does f(2)=? Am I over thinking this one? Is it just f(2)=64 also?
 
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Re: solve for f(x)a^x

goosey00 said:
If you have f(x)=a^x and f(3)=64 what does f(2)=? Am I over thinking this one? Is it just f(2)=64 also?

f(b) means evaluate f(x) at the value b. In this case f(3) means find f(x) when x=3.

Since you have the constant $a$ you can find it's value from f(3) [64 is a cube number so $a$ is a whole number] as an intermediate step. You won't need a calculator for this one but as a hint $64 = 4^3$

  1. Solve $f(3)$ in terms of $a$ by substituting $x=3$ in $f(x)=a^x$
  2. Work out the value of $a$ from the above since you're told that f(3) = 64
  3. Use that value of $a$ to find $f(2) $ by substituting $x=2$
 
Re: solve for f(x)a^x

So it would be 4^2 so f(2)=16? right?
 
Re: solve for f(x)a^x

Yes, since:

$\displaystyle a^3=4^3\,\therefore\,a=4\,\therefore\,f(x)=4^x\, \therefore\,f(2)=4^2=16$
 
Re: solve for f(x)a^x

An equivalent way to do this would be to note that a^2= (a^3)^{2/3}= 64^{2/3}.
 
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