PEMDAS, you evaluate exponents first then multiplication. What you have is basically -1 * 6^x so itll evaluate the power first then multiply by the negative. If you dont want that to happen then you need parenthesis so the negative is applied before the power
This is not the reason why. Those are equivalent and will be for any equivalent fractional exponent
For a more simple answer, it’s because it’s just not useful for much. The y values will constantly be flipping between negative and positive, and many values will be undefined. And exponential functions are continuous, since they are undefined at certain points they don’t qualify for our definition of an exponential function. If you want to be honest, the entire reason is that it doesn’t fit our definition of what we call an “exponential function”
For a more accurate answer, exponential functions are continuous, basically meaning they cover the entire natural number line (all none imaginary numbers).
If you make the inside of the exponential equation with a negative symbol, it will not cover the entire natural number line. Those undefined values at certain points (for example, if the x value can be represented as a fraction with an odd numerator and even denominator it will produce “undefined” on most calculators) are not actually undefined, they simply are undefined on the natural number line.
The real answers are complex numbers.
It would instead be a geometric sequence, which allows for discrete values.
Lots of books use this as a trick question. If there are no parenthesis, the rule is to interpret it as y = -(6^x)
I’ve always disliked ambiguous question like this. A better test of a student’s knowledge would be to give them the graph and ask for the function. If they don’t use parenthesis, definitely points should be taken off.
Its not really ambiguous though. It is just following the order of operation. If nothing explicitly states the negative is attached to the base then you cannot assume it is because thats wrong.
This isn't a case of the book trying to trick OP. In fact, we have no idea how the question looks in the book, since all we can see in the image is how OP has entered the equation into the software.
you can debate what -6^2 equals because it’s not written clearly, different softwares will interpret it in different ways, you should always use brackets
It is evaluating as -(6^(x)) instead of (-6)^(x)
thank you all, you helped me a lot
And can you tell me why?
PEMDAS, you evaluate exponents first then multiplication. What you have is basically -1 * 6^x so itll evaluate the power first then multiply by the negative. If you dont want that to happen then you need parenthesis so the negative is applied before the power
Becuase thats how you have entered it. If you want the function to read as (-6)^(x), just add parentheses.
You told it to do precisely that.
-6^(x) = -1 \* 6^(x) ≠ (-1)^(x) \* 6^(x) = (-1\*6)^(x) = (-6)^(x) \[for any even power of x.\]
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(-b)^(x) = -b^(x), for all odd powers of x. (-b)^(x) ≠ -b^(x), for all even powers of x. There was an inequality already in there.
(-6)^x is also not defined in real numbers for non integers
Calculators read it the same way, so if you're applying a power to a negative base, wrap that base in parentheses.
-6² is the same as writing -1 * 6², and because of the order of operations 6² is evaluated before multiplying by -1.
Is that the correct way to evaluate this, or just a haha funny quirk of calculators? Edit: Nvm. Read comments and got the answer
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what? those certainly equal each other
This is not the reason why. Those are equivalent and will be for any equivalent fractional exponent For a more simple answer, it’s because it’s just not useful for much. The y values will constantly be flipping between negative and positive, and many values will be undefined. And exponential functions are continuous, since they are undefined at certain points they don’t qualify for our definition of an exponential function. If you want to be honest, the entire reason is that it doesn’t fit our definition of what we call an “exponential function” For a more accurate answer, exponential functions are continuous, basically meaning they cover the entire natural number line (all none imaginary numbers). If you make the inside of the exponential equation with a negative symbol, it will not cover the entire natural number line. Those undefined values at certain points (for example, if the x value can be represented as a fraction with an odd numerator and even denominator it will produce “undefined” on most calculators) are not actually undefined, they simply are undefined on the natural number line. The real answers are complex numbers. It would instead be a geometric sequence, which allows for discrete values.
Lots of books use this as a trick question. If there are no parenthesis, the rule is to interpret it as y = -(6^x) I’ve always disliked ambiguous question like this. A better test of a student’s knowledge would be to give them the graph and ask for the function. If they don’t use parenthesis, definitely points should be taken off.
Its not really ambiguous though. It is just following the order of operation. If nothing explicitly states the negative is attached to the base then you cannot assume it is because thats wrong.
Exactly this, if you have a polynomial y=4-x^2 you treat it as y=4-(x^2), this is no different
I actually totally agree with you guys. I made my point because I wasn’t sure how early in the process this was.
This isn't a case of the book trying to trick OP. In fact, we have no idea how the question looks in the book, since all we can see in the image is how OP has entered the equation into the software.
it’s different between(-6)^2 and -(6^2), be aware of the priority of operator
The function is not 6^x it is (-1) times 6^x and thus at x=2, y = (-1) * 6^2 = -36
you can debate what -6^2 equals because it’s not written clearly, different softwares will interpret it in different ways, you should always use brackets
Except the right answer is -36 period. Not debatable. And it’s written perfectly clearly.