Notice that - x changes the sign of the input which effectively reflects the graph in the y-axis. Let's calculate some of the function's values for some strategically chosen x-coordinates.
|c|c|c|
x & 2|- x-4|+3 & f(- x)
- 2& 2|- ( -2)-4|+3 & 7
- 3& 2|- ( - 3)-4|+3 & 5
- 4& 2|- ( - 4)-4|+3 & 3
- 5& 2|- ( -5)-4|+3 & 5
- 6& 2|- ( -6)-4|+3 & 7
Now we can graph both functions by marking the points in a coordinate plane and connecting the points.
b Like in Part A, we will start by calculating some outputs for f(x). Notice that we have x in the denominator of a fraction. This means the function's domain is limited to values of x that makes the denominator nonzero.
x+4≠0 ⇔ x≠- 4
Our original function has a vertical asymptote at x=- 4 and we will chose our inputs accordingly.
|c|c|c|
[-1em]
x & 1/x+4 & f(x) [0.8em]
[-1em]
-8& 1/-8+4 & - 0.25 [0.8em]
[-1em]
-6& 1/-6+4 & - 0.5 [0.8em]
[-1em]
-5& 1/-5+4 & - 0.1 [0.8em]
[-1em]
-4.5& 1/-4.5+4 & - 2 [0.8em]
[-1em]
-3.5& 1/-3.5+4 & 2 [0.8em]
[-1em]
-3& 1/-3+4 & 0.1 [0.8em]
[-1em]
-2& 1/-2+4 & 0.5 [0.8em]
[-1em]
0& 1/0+4 & 0.25 [0.8em]
As already discussed in Part A, placing a negative before the input of a function reflects the graph in the y-axis. Let's calculate some of the function's values for a few strategically chosen x-coordinates.
|c|c|c|
[-1em]
x & 1/- x+4 & f(- x) [0.8em]
[-1em]
0& 1/- 0+4 & 0.25 [0.8em]
[-1em]
2& 1/- 2+4 & 0.5 [0.8em]
[-1em]
3& 1/- 3+4 & 0.1 [0.8em]
[-1em]
3.5& 1/- 3.5+4 & 2 [0.8em]
[-1em]
4.5& 1/- 4.5+4 & - 2 [0.8em]
[-1em]
5& 1/- 5+4 & - 0.1 [0.8em]
[-1em]
6& 1/- 6+4 & - 0.5 [0.8em]
[-1em]
8& 1/- 8+4 & - 0.25 [0.8em]
Now we can graph both functions by marking the points in a coordinate plane and connecting the points.
c let's first revise what odd and even functions are. These categories are defined as follows.
Even functions are defined as f(- x)=f(x). This means if we substitute the opposite of an input in an even function, we will get the same output for both inputs.
Odd functions are defined as f(- x)=- f(x). This means if we substitute the opposite of an input in an odd function, it will give the opposite output of f(x).
Therefore, we should find the value of each function for a pair of opposite inputs. If the outputs are neither the same or opposite, we know that the function is neither even or odd.
Part A
To find a pair of opposite inputs, we have to extend the graph somewhat in the vertical direction.
Since the outputs of 2 and -2 are neither the same nor opposite, this is neither an even or odd function.
f(- x) &≠f(x) *
f(- x) &≠- f(x) *
Part B
For this function, we will zoom in the graph from Part B.
Since the outputs of 1 and -1 are neither the same nor opposite, this is neither an even or odd function.
f(- x) &≠f(x) *
f(- x) &≠- f(x) *
