Concept Check. If ƒ(x) = a^x and ƒ(3) = 27, determine each fun...

Question

Concept Check. If ƒ(x) = a^x and ƒ(3) = 27, determine each function value. ƒ(-1)

Accepted Answer

Hello, everyone. I hope you're doing. All right. Today, today we're gonna be looking at this question that's asking us to calculate the value of the following function. If G of X is equal to two A to the exponent X and G of four is 100 and 62 then what would G of negative two be? Our answer choice is give us A is two divided by nine, B is five divided by nine C is one divided by nine and D is one divided by eight. So let's just work through this little by little first starting to see what information they tell they give us. So the first thing they tell us is that G O X is equal to two A to the exponent X. The next thing they tell us is that we're plugging in four for the X, right? So they tell us G of four. So let's write that down, we have G four is equal to two A. And now instead of being to the exponent X, it's to the exponent four. And then they tell us that when you have that G of four, the equation is equal to 100 and 62. So let's write that one more time. 100 and 62 as you go to two A to the exponent four. Now, we have all the information that we need to solve for a. So the first thing we do is divide both sides by two and that will tell us that it's 81 is equal to A to the exponent four. And to get rid of the fourth uh power, we have to take the fourth route of both sides. So it's like the fourth root of 81 is equal to the fourth route of A, to the exponent four. Now the fourth root of 81 is plus or minus three and that would equal A. Now we know that the function G of X is only defined for values of a above zero. So in this case, we're gonna stick with positive three as our answer. So that'll give us A is equal to three. And that is our answer for a. Now, with that information, we can go back into our original equation and do some more plugging in. So now our G of X equation turns into G of X is equal to two and instead of A, we plug in three, so two times three to the exponent X. And how like they're asking in the question we're trying to solve for G of negative two. So we write that as G of negative two is equal to two times three to the exponent negative two. Now, before we move forward, we can recall that when you have a negative exponent, it has a general formula of, let's say Z to the negative Y that can be rewritten as one divided by Z to the positive Y to the exponent Y right. So Z to the exponent negative Y can be rewritten as one divided by Z to the exponent positive Y. So looking at that general formula, we can rewrite the equation that we have as G of negative two is equal to two times. And since we have three to the negative two, we can rewrite that as one divided by three to the positive exponent two. Now three to the exponent two can be rewritten as just number nine. That's, that's nine. So now we would have two multiplying one divided by nine. And we know that that will give us our answer as G of negative two is equal to two divided by nine. And that will be our answer for GE of negative two which by looking at the answers choices, it is letter A. So I hope that was helpful. And until next time.

Concept Check. If ƒ(x) = a^x and ƒ(3) = 27, determine each function value. ƒ(-1)

Key Concepts

Exponential Functions

Solving for the Base in an Exponential Function

Evaluating the Function at Negative Exponents

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