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kudzordzifrancis

The general basic exponential function is of the form,

[tex]y=na^{kx} ---(1) [/tex]

The function has a [tex]y[/tex] intercept of [tex](0,3)[/tex]

This point must satisfy equation (1). So let us do the substitution to obtain,

[tex]3=na^{k(0)}[/tex]

[tex]\Rightarrow 3=n \times a^{0}[/tex]

[tex]\Rightarrow 3=n \times 1[/tex]

[tex]\Rightarrow n=3[/tex]

Now our equation 1 becomes

[tex]y=3a^{kx} ---(2)[/tex]

Also note that the graph passes through, [tex](1,1)[/tex], hence must also satisfy its equation. Substituting this point into equation (2), gives us


[tex]1=3a^{k(1)}[/tex]

[tex]\Rightarrow \frac{1}{3}=a^{k}[/tex]

Equation (2) can be rewritten as;

[tex]y=3(a^{k})^x ---(3)[/tex]

Now let us substitute the value of [tex]{a}^{k}=\frac{1}{3}[/tex] into equation (3) to obtain,

[tex]y=3(\frac{1}{3})^x [/tex]


Hence the correct answer is option B






MrRoyal

The graph illustrates an exponential function.

The function of the graph is: [tex]\mathbf{y = 3(\frac 13)^x}[/tex]

An exponential function is represented as:

[tex]\mathbf{y = ab^x}[/tex]

From the graph, we have the following points

[tex]\mathbf{(x,y) = (1,1)}[/tex]

[tex]\mathbf{(x,y) = (0,3)}[/tex]

Substitute [tex]\mathbf{(x,y) = (0,3)}[/tex] in [tex]\mathbf{y = ab^x}[/tex]

[tex]\mathbf{3 = ab^0}[/tex]

[tex]\mathbf{3 = a\times 1}[/tex]

[tex]\mathbf{3 = a}[/tex]

Rewrite as:

[tex]\mathbf{a = 3}[/tex]

Substitute [tex]\mathbf{(x,y) = (1,1)}[/tex] in [tex]\mathbf{y = ab^x}[/tex]

[tex]\mathbf{1 = ab^1}[/tex]

[tex]\mathbf{1 = ab}[/tex]

Substitute [tex]\mathbf{a = 3}[/tex]

[tex]\mathbf{1 = 3b}[/tex]

Divide both sides by 3

[tex]\mathbf{b = \frac 13}[/tex]

Substitute [tex]\mathbf{a = 3}[/tex] and [tex]\mathbf{b = \frac 13}[/tex] in [tex]\mathbf{y = ab^x}[/tex]

[tex]\mathbf{y = 3(\frac 13)^x}[/tex]

Hence, the exponential function is: [tex]\mathbf{y = 3(\frac 13)^x}[/tex]

Read more about exponential functions at:

https://brainly.com/question/11487261

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