Graph the function f ( x ) = { x if x ≤ 0 x + 1 if x > 0 f(x)=\begin{cases}x~~~~~~~~\text{if}~x\leq{0}\\x+1~\text{if}~x\gt{0}\end{cases} .

Welcome back, everyone. In this problem, we want to graph the function of F of X equals 3x minus 9 if X is less than or equal to 3, and F of X equals 3x minus 7 if X is greater than 3. And here we have a graph that will help us to plot F of X. Now notice that F of X is a piecewise function where both pieces are linear. So let's first plot the value of FFX if X is less than or equal to 3, and then if X is greater than 3. Since both of these are linear functions, that means we only need 2 points to help us plot FFX for each graph. Let's start with when X is less than our equal to 3, OK? Again, let me just keep that in red here. Now, for this, all we need is 2 points, OK? So, let's use the point at X equals 3, OK. Because no, it includes X. It includes 3 as a value for X when it says X is less than or equals to 3, and at X equals 3 F of X would be equal to 3 multiplied by 3 minus 9, which equals 0. So this tells us that 30 is a point on our graph for this condition. Next, let's use it when let's say X equals 0. We could use any other value of X, but let's use it then. Then if of 0. Would be equal to 3 minus 3 multiplied by 0 minus 9, which equals -9. So that means 09 is also on our graph. So now, let's plot 30, it should be right here. And 0-neg9. Let me use a circle instead. So, 30 and 0 negative9, OK. And now, we can connect them through a line segment, OK. Li segment would look something like this. So this is the first part of F of X, OK? Now, let's consider F of X for X greater than 3. Now when X is greater than 3 here, again we can plot it at X equals 3, but keep in mind that the point we get is going to be a hollow point since X equals 3 is not included in the subdomain of X greater than 3. So now at X equals 3, then F of we know for our function, it's gonna be 3 multiplied by 3 minus 7 because we're using the piece where X is greater than 3. That's 9 minus 7, which is 2. Therefore, 32 is going to be a hollow point on our graph. So 32, which would be right here. Now, if we choose some other point, let's say at X equals 4, OK. Then, we're gonna get F of 4 to be equal to 3 multiplied by 4 minus 7. That's 12 minus 7, which is 5. A 445 is also on our graph, and we can plot our 0.45 here. Know that we have our point, we can draw our line, OK. And if I just clean this up a bit. OK. It's clear this up a bit here on both sides. Then this would be the required graph of F of X. Thanks a lot for watching everyone. I hope this video helped.

2. Intro to Derivatives

Derivatives as Functions

Calculus

2. Intro to Derivatives

Derivatives as Functions

3:40 minutes

Problem 77aBriggs - 3rd Edition

Textbook Question

Graph the function $f (x) = {\begin{matrix} x if x \leq 0 \\ x + 1 if x > 0 \end{matrix}$ .

Verified step by step guidance

Step 1: Understand the piecewise function. The function f(x) is defined in two parts: f(x) = x for x ≤ 0 and f(x) = x + 1 for x > 0.

Step 2: Graph the first part of the function, f(x) = x, for x ≤ 0. This is a straight line through the origin with a slope of 1, but only for x-values less than or equal to 0.

Step 3: Graph the second part of the function, f(x) = x + 1, for x > 0. This is a straight line with a slope of 1, starting at the point (0, 1) and continuing for x-values greater than 0.

Step 4: Identify the point of transition at x = 0. For x = 0, the function value is 0 from the first part, so the point (0, 0) is included in the graph.

Step 5: Combine the two parts on the same set of axes. Ensure the graph is continuous at x = 0, with a closed circle at (0, 0) and an open circle at (0, 1) to indicate the transition between the two parts.

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