WEBVTT
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Use the graphs below to answer the following question.
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True or False: The equation two to the power of π₯ equals π₯ has no solution.
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Letβs begin by having a look at the two functions which we can see below.
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We might be familiar with this straight-line graph.
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Itβs the equation π¦ equals π₯, or the function π of π₯ equals π₯.
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In this function, every input value of π₯ is equal to the output value π¦.
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And so what is the other function that is also on this graph?
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The shape of this function indicates that it may be an exponential function.
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An exponential function is of the form π of π₯ equals π times π to the power of π₯, where π and π are real constants, π is positive, and π is not equal to one.
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The shape of this unknown function in blue is one of exponential growth.
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And that means that we can say that the value of π must be greater than one.
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In fact, this graph will be that of π¦ equals two to the power of π₯.
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We can check this by using some coordinates that lie on this function, for example, the coordinate one, two.
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Substituting π₯ equals one and π¦ equals two into the equation π¦ equals two to the power of π₯, we get two equals two to the power of one.
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This coordinate would lie on the function π¦ equals two to the power of π₯.
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Substituting in a different coordinate, for example, the coordinate two, four, we can see that we can verify that this coordinate will also lie on the function π¦ equals two to the power π₯.
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So how do we go from working out the two functions to establishing if two to the power of π₯ equals π₯ has any solutions or not?
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Well, we should start by remembering that the solution to an equation is the set of all values that satisfy that equation.
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Everything on the left-hand side of this equation is found on the function π¦ equals two to the power π₯.
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Everything on the right-hand side can be demonstrated by the equation or the function π¦ equals π₯.
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The point where these two are equal will be the points of intersection of the functions.
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But if we look at the graphs, we can see that there are no points of intersection.
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This means that there are no solutions to the equation two to the power π₯ equals π₯.
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And so we can give the answer that the statement βThe equation two to the power π₯ equals π₯ has no solutionβ is true.