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The functions f(x) and g(x) are shown on the graph.

The image shows two graphs. The first is f of x equals log base 3 of x and it is increasing from negative infinity in quadrant four as it goes along the y-axis and passes through 0 comma 1 to turn and increase to the right to positive infinity. The second is g of x and it is increasing from negative infinity in quadrant two as it goes along x equals negative 4 and passes through 0 comma negative 3 to turn and increase to the right to positive infinity.

Using f(x), what is the equation that represents g(x)?

g(x) = log3(x) – 4
g(x) = log3(x) + 4
g(x) = log3(x – 4)
g(x) = log3(x + 4)

The functions fx and gx are shown on the graph The image shows two graphs The first is f of x equals log base 3 of x and it is increasing from negative infinity class=

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semsee45

Answer:

[tex]g(x)=\log_3(x+4)[/tex]

Step-by-step explanation:

Translations

For [tex]a > 0[/tex]

[tex]f(x+a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units left}[/tex]

[tex]f(x-a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units right}[/tex]

[tex]f(x)+a \implies f(x) \: \textsf{translated}\:a\:\textsf{units up}[/tex]

[tex]f(x)-a \implies f(x) \: \textsf{translated}\:a\:\textsf{units down}[/tex]

Parent function:

  [tex]f(x)=\log_3x[/tex]

From inspection of the graph:

  • The x-intercept of [tex]f(x)[/tex] is (1, 0)
  • The x-intercept of [tex]g(x)[/tex] is (-3, 0)

If there was a vertical translation, the end behaviors of both g(x) and f(x) would be the same in that the both functions would be increasing from -∞ in quadrant IV.

As the x-intercepts of both functions is different, and g(x) increases from -∞ in quadrant III, this indicates that there has been a horizontal translation of 4 units to the left.

Therefore:

[tex]g(x)=f(x+4)=\log_3(x+4)[/tex]

Further Proof

Logs of zero or negative numbers are undefined.

From inspection of the graph, [tex]x=-3[/tex] is part of the domain of g(x).

Therefore, input this value of x into the answer options:

 [tex]g(-3)=\log_3(-3)-4\implies[/tex] undefined

 [tex]g(-3)=\log_3(-3)+4\implies[/tex] undefined

 [tex]g(-3)=\log_3(-3-4)=\log_3(-7)=\implies[/tex] undefined

 [tex]g(-3)=\log_3(-3+4)=\log_3(1)=0[/tex]

Hence proving that [tex]g(x)=\log_3(x+4)[/tex]

g(x) is the translated edition of f(x)

  • f(x)=log_3x

Let y=f(x)

  • y=log_3x

Now

The graph is shifted 4 units left means change in x axis.

  • y=log_3(x+4)

No change in y

So

  • g(x)=log_3(x+4)

Option D

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