# What does 2f(x) mean, in words?

1. Aug 8, 2015

### jaysquestions

Also, what about f(2x)? Im confused about how to enter 2 multiplied by the function in a ti-84 calc. Im wondering if its because Im mixing up function notation. Im learning about transforming graphs.
What does 2f(x) mean in words? Also what does f(2x) mean in words? thanks

2. Aug 8, 2015

### mathman

f(2x), value of function at point 2x.
2f(x), twice value of function at point x.

3. Aug 8, 2015

### symbolipoint

2*f(x) means two multiplied by the function f.
f(2x) means the function at 2x; or the value of the function evaluated at 2x.

Giving a name f to a function for the function using independant variable x will be named as f(x), to be read, "the function f of x". Shown alone, f and x are not factors, but are a complete name.

4. Aug 8, 2015

### jaysquestions

thanks for the answers , so how would I enter 2f(x) into a calc then? for example, f(x) = x^2, .. how do I enter two times the function into calc?

5. Aug 8, 2015

### jaysquestions

The attached graph from textbook is what is confusing me. I don't understand the reason for the vertical shift?

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6. Aug 8, 2015

### Staff: Mentor

Assuming that you have the graph of y = f(x), a new function y = 2*f(x) has y values that are exactly twice the value of those on the graph of y = f(x). This can be thought of as an expansion of the graph of f away from the x-axis to get y = 2*f(x). Don't think of this transformation as a shift, which is a transformation that rigidly moves all of the points on one graph by a set amount.

The graph of y = f(x) + 3 shifts (or translates) all of the points on the graph of f by 3 units upward. For example, if (2, 5) is on the graph of f, then (2, 8) will be on the shifted, or translated, graph.

7. Aug 8, 2015

### jaysquestions

Sorry im still not understanding and thanks for the reply. Its figure 1.36 that is giving me problem. For example, if y = f(x) = x^2, how would the new function 2f(x) = x^2 transform the function downward, as the graph is showing? I think Im assuming something wrong about the graph but I cant see what my error is ? thank you

8. Aug 8, 2015

### SteamKing

Staff Emeritus
Let's say that the function f(x) = -1 when x = 0. What is y = 2f(x) when x = 0? How would that look plotted on the same axes as y = f(x)?

9. Aug 8, 2015

### symbolipoint

A factor on the function either STRETCHES or SHRINKS it vertically. This is not a movement of the function from one place to another place. The figures 1.36, 1.37, 1.38, 1.39, are the book's attempt to show this using examples.

10. Aug 8, 2015

### jaysquestions

When it shrinks or stretches does the vertex stay in the same place though? For example, does f(x) = x^2 and 2f(x) = x^2 have the same vertex coordinates? thanks

11. Aug 8, 2015

### Staff: Mentor

You're misinterpreting the graph. Each y value on the graph of y = x2 is doubled to get the graph of y = 2f(x). Part of the black graph (y = f(x)) lies below the y-axis, so the points on the red graph (y = 2f(x)) are twice as far below the x-axis as those on the black graph. All points on the red graph are twice as far away from the x-axis as those on the black graph.

Also, if f(x) = x2, then 2f(x) = 2x2, not x2 as you wrote.

Again, 2f(x) would be 2x2, not x2. For y = x2, the vertex is at (0, 0), so doubling the y-value has no effect. The graph of y = 2x2 will also have (0, 0) as its vertex. All other points will be twice as far away from the x-axis as those on the graph of y = x2.

12. Aug 8, 2015

### DrewD

The example given does not use $f(x)=x^2$. Lets assume that it uses $g(x)=x^2-1$. Then the vertex is at $(0,-1)$. If $2f(x)$ means that the $y$ value of each point is multiplied by two, would the vertex stay in the same place?

13. Aug 8, 2015

### jaysquestions

OK Im getting this now. I was making two strange assumptions: 1) that the function represented in fig 1.36 as y=f(x) is f(x) = x^2. (it cant be because the vertex is well below the origin) and 2) I was thinking parabolas cant have vertexes below x-axis because of the x^2, but obviously they can be shifted, which is what fig 1.36 is and the whole point of all the examples. So in answer to your g(x) = x^2 -1 example, if we use 2f(x) then vertex would not stay in same place in your example, it would shift.
thank you

14. Aug 9, 2015

### jaysquestions

I want to make sure Im understanding the diff between 2f(x) and f(2x) using an example. So if, f(x) = x2 - 3, am I correct in saying that:

1) 2f(x) = 2(x2-3)?
and
2) f(2x) = 2x2-3? or should this be = (2x2)-3?

thanks

15. Aug 9, 2015

### SteamKing

Staff Emeritus
1) 2f(x) = 2(x2-3) is correct.

2) f(2x) = 2x2-3? or should this be = (2x2)-3?
Neither.

f(x) = x2 - 3, therefore f(2x) = (2x)2 - 3 = 4x2 - 3

Look at it like this. Take a function f(t) = t2 - 3 and set t = 2x.

16. Aug 9, 2015

### Svein

Neither. Let us take it step by step. First $f(x)=x^{2}-3$. This also means that $f(u)=u^{2}-3$ (we only changed the name of the variable, nothing else). So if we substitute 2x for u, we get $f(2x)=(2x)^{2}-3$.

17. Aug 9, 2015

### jaysquestions

Thanks for the replies , ..believe it or not I actually meant to say f(2x) = (2x)2-3, I just overlooked where the bracket started.
thanks everyone for the help