Video Transcript
The function 𝑓 of 𝑥 is represented by the line 𝐴𝐵 and the function 𝑔 of 𝑥 is represented by the line 𝑂𝐴 where the coordinates of 𝐴 are four, two. Find 𝑓 of negative eight plus 𝑔 of negative 10.
Let’s think about the information we’re given. 𝑓 of 𝑥 is the line 𝐴𝐵, so we can label that. 𝑔 of 𝑥 is represented by the line 𝑂𝐴, which is this line. We also know that 𝐴 is located at the point four, two. Using the coordinates of 𝐴 four, two, we can find the coordinates of 𝐵. We know that 𝐵 is located at zero along the 𝑥-axis since it intersects the 𝑦-axis. And since 𝐴 and 𝐵 fall along the same line, which is a horizontal line, the 𝑦-coordinates will be the same. And we can say that 𝐵 is located at zero, two. Our final point is 𝑂, which is located at the origin zero, zero.
In order for us to add 𝑓 of negative eight and 𝑔 of negative 10, we’ll need to find out what the functions for 𝑓 of 𝑥 and 𝑔 of 𝑥 are. If we think about the form 𝑦 equals 𝑚𝑥 plus 𝑏, we can use that to help us write an equation for the function 𝑓 of 𝑥. The first thing we need is a slope. Remember that 𝑓 of 𝑥 goes through the points 𝐴 and 𝐵. When we have two points, we can find the slope by subtracting 𝑦 two minus 𝑦 one and putting that over 𝑥 two minus 𝑥 one.
In the case of 𝐴 and 𝐵, this will be two minus two over four minus zero. But two minus two is zero, and that means our slope is equal to zero. This makes sense to us because horizontal lines always have a slope of zero. This means the first part of our function will be zero times 𝑥. And the 𝑏 portion of our formula represents the 𝑦-intercept. In this case, the line crosses the 𝑦-axis at point 𝐵, which is located at two along the 𝑦-axis. That means we can say 𝑓 of 𝑥 equals zero times 𝑥 plus two, which, when we simplify, is just 𝑓 of 𝑥 equal to two.
We want to follow the same procedure for 𝑔 of 𝑥. 𝑔 of 𝑥 is the line that crosses the points 𝐴 and 𝑂. We can use these two points to find the slope, which will be two minus zero over four minus zero, two over four, or one-half. The first part of our function for 𝑔 of 𝑥 is then one-half times 𝑥. And then we need to consider 𝐵, the place where this line crosses the 𝑦-axis. This line goes through the origin. And therefore, the 𝑏-value is zero. So we have 𝑔 of 𝑥 equals one-half 𝑥 plus zero, which we can simplify to one-half 𝑥.
Now that we have the formulas for each of these functions, we can add 𝑓 of negative eight and 𝑔 of negative 10. Now for our 𝑓 of 𝑥 function, it doesn’t matter what 𝑥 is, the output will always be two. This is the case for horizontal lines. So 𝑓 of negative eight is going to be equal to two. There’s no 𝑥-value to plug in. However, when we look at our 𝑔 of 𝑥 formula, we do need to plug in negative 10 for 𝑥.
Since 𝑔 of 𝑥 is equal to one-half times 𝑥, 𝑔 of negative 10 will be equal to one-half times negative 10. One-half times negative 10 is negative five. So we have two plus negative five, which is negative three. Under all of these conditions, 𝑓 of negative eight plus 𝑔 of negative 10 is equal to negative three.
