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MrMuchimi
2 years ago
10

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

Mathematics
2 answers:
katrin [286]2 years ago
7 0

Answer:

g(x)=\log_3(x+4)

Step-by-step explanation:

<u>Translations</u>

For a > 0

f(x+a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units left}

f(x-a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units right}

f(x)+a \implies f(x) \: \textsf{translated}\:a\:\textsf{units up}

f(x)-a \implies f(x) \: \textsf{translated}\:a\:\textsf{units down}

<u>Parent function:</u>

  f(x)=\log_3x

From inspection of the graph:

  • The x-intercept of f(x) is (1, 0)
  • The x-intercept of g(x) 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 <u>horizontal translation</u> of <u>4 units to the left</u>.

Therefore:

g(x)=f(x+4)=\log_3(x+4)

<u>Further Proof</u>

Logs of zero or negative numbers are <u>undefined</u>.

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

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

 g(-3)=\log_3(-3)-4\implies undefined

 g(-3)=\log_3(-3)+4\implies undefined

 g(-3)=\log_3(-3-4)=\log_3(-7)=\implies undefined

 g(-3)=\log_3(-3+4)=\log_3(1)=0

Hence proving that g(x)=\log_3(x+4)

Dafna11 [192]2 years ago
7 0

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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