Answered

The function [tex]f(x)=0.11(3)^x[/tex] is reflected over the [tex]x[/tex]-axis to produce function [tex]g(x)[/tex]. Function [tex]g(x)[/tex] is then reflected over the [tex]y[/tex]-axis to produce function [tex]h(x)[/tex]. Which function represents [tex]h(x)[/tex]?

A. [tex]h(x)=-0.11(3)^x[/tex]
B. [tex]h(x)=0.11(3)^{-x}[/tex]
C. [tex]h(x)=0.11(3)^x[/tex]
D. [tex]h(x)=-0.11(3)^{-x}[/tex]



Answer :

GinnyAnswer
Let's start with the initial function given:

[tex]\[ f(x) = 0.11 \cdot 3^x \][/tex]

### Step 1: Reflecting [tex]\( f(x) \)[/tex] over the [tex]\( x \)[/tex]-axis to obtain [tex]\( g(x) \)[/tex]

Reflecting a function over the [tex]\( x \)[/tex]-axis involves multiplying the function by [tex]\(-1\)[/tex]. Therefore, the function [tex]\( g(x) \)[/tex] after reflection over the [tex]\( x \)[/tex]-axis will be:

[tex]\[ g(x) = -f(x) = -0.11 \cdot 3^x \][/tex]

### Step 2: Reflecting [tex]\( g(x) \)[/tex] over the [tex]\( y \)[/tex]-axis to obtain [tex]\( h(x) \)[/tex]

Reflecting a function over the [tex]\( y \)[/tex]-axis involves replacing [tex]\( x \)[/tex] with [tex]\(-x\)[/tex]. Applying this transformation to [tex]\( g(x) \)[/tex]:

[tex]\[ h(x) = g(-x) \][/tex]

Substitute [tex]\(-x\)[/tex] into the expression for [tex]\( g(x) \)[/tex]:

[tex]\[ h(x) = -0.11 \cdot 3^{-x} \][/tex]

Thus, the function [tex]\( h(x) \)[/tex] after reflecting [tex]\( g(x) \)[/tex] over the [tex]\( y \)[/tex]-axis is:

[tex]\[ h(x) = -0.11 \cdot 3^{-x} \][/tex]

### Conclusion
After performing the reflections as described, the function [tex]\( h(x) \)[/tex] is:

[tex]\[ h(x) = -0.11 \cdot 3^{-x} \][/tex]

Among the given options, this corresponds to:

[tex]\[ \boxed{h(x) = -0.11 \cdot 3^{-x}} \][/tex]

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