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.