Describe these functions

Question

Describe $f(-x), -f(x), f(10-x),-f(10-x)$ with respect to function $f(x)$.

Taking any example please how these functions can plot in a graph

Say, $f(10-x)=f(-(x-10))$ right?

Then how it is different from $-f(10-x)=-f(-(x-10))$ if we plot it as a simple graph?

Answer 1

Suppose $f(x)=x^2+3x$

$$f(-x)=(-x)^2+3(-x)=x^2-3x$$ All this does is flips the graph in the $y$-axis. You can see that here

$$-f(x)=-[x^2+3x]=-x^2-3x$$ This flips the graph in the $x$-axis, seen here $$f(10-x)=f(-x+10)$$ $$f(-x+10)=(-x+10)^2+3(-x+10)=x^2-20x+100-3x+30=x^2-23x+130$$

This takes the graph of $f(-x)$ and shifts it along the $x$-axis by $10$, seen here $$-f(-x+10)=-[x^2-23x+130]=-x^2+23x-130$$

This flips the graph of $f(-x+10)$ in the $x$-axis, seen here.

Hope this helps.

Answer 2

$f\left(-x\right)$ is doing a symmetry to the graph of $f$ with respect to the $y$ axis while $f(-x)$ is doing the same but respect to the $x$ axis. $f(10-x)$ is to do the symmetry respectively to the $y$ axis but then translate it from $10$ unit to the left. I let you do the last one

Answer 3

Given the graph of y= f(x) then the graph of y= f(-x) is just that graph of y= f(x) reflected in the y-axis. The point (x, y) becomes (-x, y). The graph of y= -f(x) is the graph of y= f(x) reflected in the x-axis. The point (x, y) becomes (x, -y).

Yes, f(10- x)= f(-(x- 10). The graph of y= f(x- 10) is just the graph of y= f(x) translated, along the x-axis, to the right a distance 10. The point (x, y) becomes (x- 10, y). So the graph of y= f(10- x) is the graph of y= f(x) first shifted to the right a distance 10 then reflected in the y-axis.

Answer 4

It may be helpful to think about a function as something that maps input values to output values.

Given the graph of $f(x)$,

$f(-x)$ can be thought of as swapping the output for each $x$ input with that of the negative input $-x$. So the point $(4,8)$ becomes $(-4,8)$ etc. This is the same as flipping $f(x)$ across the $y$-axis.

$-f(x)$ means you reverse the sign of each output $y$-value. So $6$ becomes $-6$, etc. This is the same as flipping the graph of $f(x)$ across the $x$-axis.

$f(10-x)$ is a reflection of $f(x)$ in the line $x=5$. This is a bit harder to grasp, but think of it this way:

• When $x=5$, $f(10-5)=f(5)$, so the graph is unchanged here.
• When $x=6$, $f(10-6)=f(4)$, so you swap around the $y$-values of $x=4$ and $x=6$.
• When $x=7$, $f(10-7)=f(3)$, so you swap around the $y$-values of $x=3$ and $x=7$.
• When $x=8$, $f(10-8)=f(2)$, so you swap around the $y$-values of $x=2$ and $x=8$.

Do you see a pattern?

(Speaking generally, the graph of $f(2a-x)$ is the graph of $f(x)$ reflected across the line $x=a$.)

Finally, $-f(10-x)$ is the reflection of $f(10-x)$ across the $x$-axis.

Answer 5

• The graph of $f(-x)$ is the set of points $(-x, f(x))$ which are the symmetric points of $(x,f(x))$ w.r.t. the $y$-axis.
• Similarly, the graph of $-f(x)$ is symmetric of the graph of $f(x)$ w.r.t. the $x$-axis.
• The graph of $f(10-x)$ is symmetric of the graph of $f(x)$ w.r.t. the vertical axis with equation $x=5$.
• The graph of $-f(10-x)$ is obtained composing transformation n°3 by symmetry n°2, hence we obtain the symmetry w.r.t. the point $(5,0)$.