 namenerk

2021-12-18

What is the difference between f(-x) and -f(x)?
What is the difference between in terms of their graphs? Jenny Bolton

The graph of f(-x) is the mirror image of the graph of f(x) with respect to the vertical axis.
The graph of -f(x) is the mirror image of the graph of f(x) with respect to the horizontal axis.
A function is called even if $f\left(x\right)=f\left(-x\right)$ for all x (For example, cos(x)).
A function is called odd if $-f\left(x\right)=f\left(-x\right)$ for all x (For example, sin(x)). Linda Birchfield

The most helpful vocabulary related to your question has to do with the parity of a given function. Functions are described as odd, even, neither. Most functions are neither odd nor even, but it is great to know which ones are even or odd and how to tell the difference.
Even functions - If f(x) is an even function, then for every x and -x in the domain of f, $f\left(x\right)=f\left(-x\right)$. Graphically, this means that the function is symmetric with respect to the y-axis. Thus, reflections across the y-axis do not affect the function's appearance. Good examples of even functions include: ${x}^{2},{x}^{4},\dots ,{x}^{2n}$ (integer n); cos(x), cosh(x), and |x|.
Odd functions - If f(x) is an odd function, then for every x and −x in the domain of f, $-f\left(x\right)=f\left(-x\right)$. Graphically, this means that the function is rotationally symmetric with respect to the origin. Thus, rotations of ${180}^{\circ }$ or any multiple of ${180}^{\circ }$ do not affect the function's appearance. Good examples of odd functions include: $x,{x}^{3},{x}^{5},\dots ,{x}^{2n+1}$ (integer n); sin(x), and sinh(x).
To really get a feel for recognizing these functions, I suggest you graph several (if not all) of them. You will be well-equipped to visually determine the parity of most functions if you just spend a little time graphing the example functions. RizerMix

Result:
- $f\left(-x\right)$ reflects the graph of $f\left(x\right)$ across the $y$-axis.
- $-f\left(x\right)$ reflects the graph of $f\left(x\right)$ across the $x$-axis.
Solution:
The difference between $f\left(-x\right)$ and $-f\left(x\right)$ is that $f\left(-x\right)$ represents the function $f$ evaluated at the negation of $x$, whereas $-f\left(x\right)$ represents the negation of the function $f$ evaluated at $x$.
In terms of their graphs, $f\left(-x\right)$ is obtained by reflecting the graph of $f\left(x\right)$ across the $y$-axis. This means that any point $\left(x,y\right)$ on the graph of $f\left(x\right)$ is transformed to $\left(-x,y\right)$ on the graph of $f\left(-x\right)$.
On the other hand, $-f\left(x\right)$ is obtained by taking the opposite (negation) of the $y$-values of the graph of $f\left(x\right)$. This means that every point $\left(x,y\right)$ on the graph of $f\left(x\right)$ is transformed to $\left(x,-y\right)$ on the graph of $-f\left(x\right)$. Vasquez

Step 1. Algebraic Difference:
The expression $f\left(-x\right)$ represents the function $f\left(x\right)$ evaluated at the input $-x$. It means that we substitute $-x$ for $x$ in the original function $f\left(x\right)$.
On the other hand, $-f\left(x\right)$ represents the negation of the function $f\left(x\right)$. It means that we take the opposite of the output of the function for any given $x$.
Step 2. Graphical Difference:
The graph of $f\left(-x\right)$ is obtained by reflecting the graph of $f\left(x\right)$ about the y-axis. This reflection is due to the substitution $-x$ in place of $x$.
The graph of $-f\left(x\right)$ is obtained by reflecting the graph of $f\left(x\right)$ about the x-axis. This reflection is due to the negation of the function's outputs.
In summary, the difference between $f\left(-x\right)$ and $-f\left(x\right)$ is twofold: algebraically, the former substitutes $-x$ into the function, while the latter negates the function's outputs. Graphically, $f\left(-x\right)$ reflects the graph about the y-axis, while $-f\left(x\right)$ reflects the graph about the x-axis. Don Sumner

To solve the problem, let's start by understanding the difference between $f\left(-x\right)$ and $-f\left(x\right)$ algebraically and then examine their graphical differences.
1. Algebraic Difference:
The expression $f\left(-x\right)$ represents the function $f$ evaluated at $-x$, whereas $-f\left(x\right)$ represents the negative of the function $f$ evaluated at $x$.
2. Graphical Difference:
To understand the graphical differences, let's consider the following cases:
Case 1: If $f\left(x\right)$ is an even function, meaning $f\left(x\right)=f\left(-x\right)$ for all $x$ in the domain of $f$, then $f\left(-x\right)$ and $-f\left(x\right)$ would have identical graphs. This is because substituting $-x$ into an even function does not change its value, and taking the negative of an even function results in the same function.
Case 2: If $f\left(x\right)$ is an odd function, meaning $f\left(x\right)=-f\left(-x\right)$ for all $x$ in the domain of $f$, then $f\left(-x\right)$ and $-f\left(x\right)$ would have graphs that are symmetric with respect to the origin. Substituting $-x$ into an odd function changes the sign of the function, and taking the negative of an odd function again changes the sign, resulting in a graph symmetric about the origin.
Case 3: If $f\left(x\right)$ is neither even nor odd, $f\left(-x\right)$ and $-f\left(x\right)$ would generally have different graphs. Substituting $-x$ into the function and taking the negative of the function would lead to different values, which results in different graphical representations.
In summary, the difference between $f\left(-x\right)$ and $-f\left(x\right)$ lies in their algebraic expressions and their corresponding graphical representations. The algebraic difference depends on the properties of the original function $f\left(x\right)$, whether it is even, odd, or neither. The graphical difference can vary based on the symmetry properties of the function and how they change when $-x$ is substituted or when the negative sign is applied.

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