Question Video: Solving Exponential Equations Graphically Mathematics

The diagram shows the graph of 𝑓(π‘₯) = 2^(π‘₯/2). Use this graph to find the solution set of the equation 2^(π‘₯/2) + 5 = 9.

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Video Transcript

The diagram shows the graph of 𝑓 of π‘₯ is equal to two to the power of π‘₯ over two. Use this graph to find the solution set of the equation two to the power of π‘₯ over two plus five is equal to nine.

In this question, we’re given the graph of an exponential function 𝑓 of π‘₯. And we’re asked to use this to determine the solution set of an equation which contains our function 𝑓 of π‘₯. To do this, we start by recalling the solution set of an equation is the set of all solutions to that equation. In this case, it will be the set of all values of π‘₯, such that two to the power of π‘₯ over two plus five is equal to nine. To answer this question, it can help us to rewrite our exponential equation in terms of the function 𝑓 of π‘₯. Substituting two to the power of π‘₯ over two is equal to 𝑓 of π‘₯ into our equation, we get 𝑓 of π‘₯ plus five is equal to nine. We can simplify this equation further by subtracting five from both sides. We get 𝑓 of π‘₯ is equal to nine minus five, which simplifies to give us 𝑓 of π‘₯ is equal to four. So, we want to find the values of π‘₯ such that our function outputs a value of four.

Remember that the 𝑦-coordinate of any point on our curve tells us the output value of our function at that value of π‘₯. So, we want to find all of the points on our curve with 𝑦-coordinate four. We do this by sketching the line 𝑦 is equal to four onto our diagram. We can see there’s only one point of intersection between our line and our curve. And we can see that this point has π‘₯-coordinate four. Therefore, when we input a value of π‘₯ is equal to four into our function, the output value is four. 𝑓 of four is equal to four. And in fact, since this is the only point of intersection between our line and our curve, this is the only solution to our equation. Therefore, the solution set of this equation is the set containing four.

We can check that π‘₯ is equal to four is a solution to our equation by substituting π‘₯ is equal to four into the left-hand side of our equation. Substituting π‘₯ is equal to four into the left-hand side of our equation, we get two to the power four over two plus five, which we can simplify four over two is equal to two. So, this is equal to two squared plus five. And then we can evaluate this. Two squared is equal to four. So, we get four plus five, which is equal to nine, which we can see is exactly equal to the right-hand side of this equation. Therefore, four is a solution to our equation, and we know it’s the only solution. Therefore, the solution set of the equation two to the power of π‘₯ over two plus five is equal to nine is the set containing four.

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