a. Reflecting the graph vertically means that each output value will be reflected over the horizontal [latex]t[/latex]–axis as shown below.

Vertical reflection of the square root function

Because each output value is the opposite of the original output value, we can write

[latex]V\left(t\right)=-s\left(t\right)\text{ or }V\left(t\right)=-\sqrt{t}[/latex]

Notice that this is an outside change, or vertical shift, that affects the output [latex]s\left(t\right)[/latex] values, so the negative sign belongs outside of the function.

b. Reflecting horizontally means that each input value will be reflected over the vertical axis as shown below.

Horizontal reflection of the square root function

Because each input value is the opposite of the original input value, we can write

[latex]H\left(t\right)=s\left(-t\right)\text{ or }H\left(t\right)=\sqrt{-t}[/latex]

Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.

Note that these transformations can affect the domain and range of the functions. While the original square root function has domain [latex]\left[0,\infty \right)[/latex] and range [latex]\left[0,\infty \right)[/latex], the vertical reflection gives the [latex]V\left(t\right)[/latex] function the range [latex]\left(-\infty ,0\right][/latex] and the horizontal reflection gives the [latex]H\left(t\right)[/latex] function the domain [latex]\left(-\infty ,0\right][/latex].

1. For [latex]g\left(x\right)[/latex], the negative sign outside the function indicates a vertical reflection, so the [latex]x[/latex]-values stay the same and each output value will be the opposite of the original output value.
   

   [latex]x[/latex]	2	4	6	8
   [latex]g\left(x\right)[/latex]	–1	–3	–7	–11

2. For [latex]h\left(x\right)[/latex], the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the [latex]h\left(x\right)[/latex] values stay the same as the [latex]f\left(x\right)[/latex] values.
   

   [latex]x[/latex]	−2	−4	−6	−8
   [latex]h\left(x\right)[/latex]	1	3	7	11

Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let’s begin with the rule for even functions.

[latex]f\left(-x\right)={\left(-x\right)}^{3}+2\left(-x\right)=-{x}^{3}-2x[/latex]

This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.

[latex]-f\left(-x\right)=-\left(-{x}^{3}-2x\right)={x}^{3}+2x[/latex]

Because [latex]-f\left(-x\right)=f\left(x\right)[/latex], this is an odd function.

Analysis of the Solution

Consider the graph of [latex]f[/latex]. Notice that the graph is symmetric about the origin. For every point [latex]\left(x,y\right)[/latex] on the graph, the corresponding point [latex]\left(-x,-y\right)[/latex] is also on the graph. For example, (1, 3) is on the graph of [latex]f[/latex], and the corresponding point [latex]\left(-1,-3\right)[/latex] is also on the graph.