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Graphing Exponential Equations

For each function, determine the domain, range, intercepts, asymptotes, end behavior, and
where the function is increasing or decreasing. Then sketch the graph.

F(x)=1/4*(1/7)(^x-1)+2

User Reyaner
by
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1 Answer

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Answer:

To analyze the function f(x) = (1/4) * (1/7)^(x-1) + 2, let's determine its domain, range, intercepts, asymptotes, and behavior, and then sketch the graph.

Domain:

The function is defined for all real numbers.

Range:

Since (1/7)^(x-1) is always positive or zero, the range of the function is [2, ∞), meaning it takes on values starting from 2 and goes to positive infinity.

Intercepts:

To find the x-intercept, we set f(x) = 0 and solve for x:

(1/4) * (1/7)^(x-1) + 2 = 0

(1/4) * (1/7)^(x-1) = -2

Since (1/7)^(x-1) is always positive or zero, there are no x-intercepts.

To find the y-intercept, we substitute x = 0 into the function:

f(0) = (1/4) * (1/7)^(0-1) + 2

f(0) = (1/4) * 7 + 2

f(0) = 7/4 + 2

f(0) = 15/4

The y-intercept is (0, 15/4) or (0, 3.75).

Asymptotes:

The function has a horizontal asymptote at y = 2, as x approaches positive or negative infinity.

Increasing or Decreasing:

The function is decreasing as x increases.

End Behavior:

As x approaches positive infinity, the function approaches the horizontal asymptote y = 2.

Now, let's sketch the graph of the function:

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The graph of the function starts at the y-intercept (0, 3.75), decreases as x increases, and approaches the horizontal asymptote y = 2 as x goes to positive infinity.

Please note that the sketch is a rough representation and may not be perfectly accurate.

User Vitaliy Prushak
by
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