The given figure shows the graphs of the function 𝑓 of 𝑥 equals five 𝑥 minus four and 𝑔 of 𝑥 equals negative 𝑥 plus eight. What is the point where 𝑓 of 𝑥 is equal to 𝑔 of 𝑥?
Our question is asking us where is 𝑓 of 𝑥 equal to 𝑔 of 𝑥. This will be the intersection of the two functions. We also might call it a solution. A solution is the coordinate 𝑥, 𝑦 such that that value for 𝑥 and 𝑦 satisfies both equations.
Now in our case, we’ve been given a graph. And we see that the intersection is here. It’s located at two along the 𝑥-axis and six along the 𝑦-axis. Our intersection will be an 𝑥- and then a 𝑦-coordinate. Two, six is the intersection here. It’s the place where the function 𝑓 of 𝑥 and the function 𝑔 of 𝑥 are equal to each other. This is solving the problem graphically.
But we can also take these two functions and solve them algebraically. If we want to find the place where 𝑓 of 𝑥 is equal to 𝑔 of 𝑥, we substitute five 𝑥 minus four in for 𝑓 of 𝑥 and negative 𝑥 plus eight in for 𝑔 of 𝑥. And then we solve for 𝑥. To get the 𝑥s on the same side, we can add 𝑥 to both sides. Five 𝑥 plus 𝑥 equals six 𝑥, and negative 𝑥 plus 𝑥 equals zero. So the right-hand side is left with only eight.
From there, we add four to both sides, and we get six 𝑥 equals 12. We divide both sides by six. 12 divided by six is two, so 𝑥 equals two. This is saying that when 𝑥 equals two, these two functions are equal. And we’ve already seen that on the graph.
Let’s go back to the top and plug in two for our 𝑥 values. Is five times two minus four equal to negative two plus eight? Five times two is 10, minus four is six. Negative two plus eight also equals six. And this gives us the 𝑦-value where our function is equal to each other. When 𝑥 equals two, our 𝑓 of 𝑥 and our 𝑔 of 𝑥 equal six, which confirms the intersection or the solution two, six.