r/learnmath • u/Lookinglikeamushroom New User • 17h ago

How would you find x when comparing an exponential function with a linear one?

If I had an exponential function called:

g(x) = c^x + d

and a linear function called:

f(x) = ax+b

Besides just putting them into a graph, how would I find the points where they cross?

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u/Brightlinger MS in Math 10h ago

Practically speaking, there's no "nice" way to solve this algebraically by hand. However, if you see this problem in a book, often it will be carefully chosen to have a nice solution, and can be solved "by inspection", meaning you just look at it and guess the solution because it's a small whole number.

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u/Gold_Ad8890 New User 14h ago

the function f is the set of all points (x, f(x)). the function g is the set of all points (x, g(x)). to intersect, they must contain the same point, that is (x, f(x)) = (x, g(x)). this is true if and only if x = x and f(x) = g(x). thus it is true if and only if f(x) = g(x). so just set f(x) = g(x) and solve the equation for x.

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u/FormulaDriven Actuary / ex-Maths teacher 13h ago

That's a detailed explanation of how to express the problem as an equation to be solved - unfortunately, it's the solving of

cx + d = ax + b

that is the hard part. A solution can be expressed in terms of the Lambert W function, but that could be challenging computationally. OP might be better off using a numerical method such as iteration to solve the equation.

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u/Gold_Ad8890 New User 13h ago

yes, i'm answering the question op asked. if op meant a different question, it is their responsibility to inform me of that, not my responsibility to guess it.

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u/FormulaDriven Actuary / ex-Maths teacher 13h ago

You have made a guess though: that the OP doesn't know the concept of equating two functions to find where their graphs cross, but is perfectly comfortable solving an equation which doesn't have neat algebraic solution. Both those steps have to be accomplished to answer the question the OP wants to ask, so I think it is you who has chosen to make an unusual interpretation of the question as stated.

You saying "just set f(x) = g(x) and solve..." doesn't even acknowledge that might require a bit of work given that the OP has stated the two functions he is dealing with.

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u/Gold_Ad8890 New User 13h ago

no, if i've made an "assumption", and i don't think i have because i think it's empirically true, it's that an explanation should begin at the first point op hasn't directly communicated they understand and wait for confirmation from op before continuing.

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u/FormulaDriven Actuary / ex-Maths teacher 13h ago

OK - I don't see this as a big thing to disagree about.

All I really meant to highlight in my original reply to you was that it was worth at least acknowledging that the hard part is solving that equation. I actually agree with you that from that point we should await the OP coming back to ask more about techniques for doing that.

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u/chkntendis Physicist 8h ago

That’s the general idea but this is asking for specifically when f is an exponential function and g is a linear one. I think that if such a question is asked, the basic method is known. At least for me, that method came before exponential functions. And in that specific case you don’t have a simple (closed form) solution for all cases.