Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

If y=2^x is an exponential function and y=3^x is an exponential function, then how about ...?

y=2^x + 3^x. Is it still an exponential function? Why or why not?

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  • 1 month ago
    Favourite answer

    An exponential function has the following form:

    y = ab^x, where b is a positive real number (but b ≠ 1)

    y = 2^x is an exponential function with a = 1, b = 2

    y = 3^x is an exponential function with a = 1, b = 3

    If y = 2^x + 3^x was an exponential function, we should be able to rewrite it as:

    y = ab^x

    Let's assume that it *is* possible to do so. Can we find values of a and b that work?

    When x = 0, then we have:

    y = 2^0 + 3^0 = 1 + 1 = 2

    Mapping that to y = ab^0 = 2 that means a = 2

    y = 2b^x

    And when x = 1, then we have:

    y = 2^1 + 3^1 = 5 = 2b^1

    y = 2b = 5

    b = 5/2

    However, that's where things break down.

    y = 2(5/2)^x

    That does work for the two points we've chosen (x=0 and x=1), but it doesn't work elsewhere.

    For example, x = -1.

    y = 2^-1 + 3^-1 = 1/2 + 1/3 = 3/6 + 2/6 = 5/6

    But with our new exponential function:

    y = 2(5/2)^-1

    y = 2 * (2/5)

    y = 4/5

    Answer:

    No, clearly you can't make the sum of those two exponential function into a single exponential function.

    2^x + 3^x is not in the correct form y = ab^x

  • ?
    Lv 7
    1 month ago

    We know that y₁ = 2ˣ  and y₂ = 3ˣ are both exponential functions.

    y₃ = 2ˣ + 3ˣ is also an exponential function because

    .....for each value x, y₃ =  y₁ + y₂

    .....Example. When x = 0, y₁ = y₂ = 1 and y₃ = 1 + 1 = 2

    .....Example. When x = -1, y₁ = 0.333, y₂ = 0.5 and y₃ = 0.333 + 0.5 = 0.833

    Additionally, the graph of y₃ = 2ˣ + 3ˣ confirms it is an exponential function ......ANS

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  • rotchm
    Lv 7
    1 month ago

    First, you need to know what the words mean. You need to know what "exponential function" means. 

    So, go find its definition, and state it here. 

    Then, if there is anything in that definition you don't understand, say so.

    Then we will take it from there. 

  • Anonymous
    1 month ago

    Sorry, my answer is "Anonymous" as your post is Anonymous. There is no need for an Anonymous question in this category, Most 'Anonymous' posts are trolls, maybe yours is not, but the high likelihood is there. If you post as yourself, I would be glad to help you with any problem you have.

    I spend a lot of time on each problem, I just want to make sure it's for a worthwhile cause.

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  • fcas80
    Lv 7
    1 month ago

    Is there some number a so that 2^x + 3^x = a^x?  I don't think so.

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