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Difficulty: MediumFunction Notation and Transformations

The graph of the function y=f(x)y = f(x) in the xyxy-plane passes through the point (3,−4)(3, -4). If the function gg is defined by g(x)=f(x+2)−1g(x) = f(x + 2) - 1, which of the following points must lie on the graph of y=g(x)y = g(x)?

  1. (1,−5)(1, -5)Answer
  2. B
    (5,−5)(5, -5)
  3. C
    (1,−3)(1, -3)
  4. D
    (5,−3)(5, -3)

Answer

(1,−5)(1, -5)
The correct answer is the point (1,−5)(1, -5). Since the point (3,−4)(3, -4) lies on the graph of ff, we know that f(3)=−4f(3) = -4. For the function g(x)=f(x+2)−1g(x) = f(x + 2) - 1, substituting x=1x = 1 gives g(1)=f(1+2)−1=f(3)−1g(1) = f(1 + 2) - 1 = f(3) - 1. Substituting −4-4 for f(3)f(3) yields g(1)=−4−1=−5g(1) = -4 - 1 = -5. Thus, the point (1,−5)(1, -5) must lie on the graph of gg.

Step-by-Step Solution

1
Identify the given function value from the point on the graph of ff.
Since the graph of y=f(x)y = f(x) passes through (3,−4)(3, -4), we have f(3)=−4f(3) = -4.
Any point (x,y)(x, y) on the graph of a function satisfies the equation y=f(x)y = f(x).
2
Determine the input variable xx for the function g(x)=f(x+2)−1g(x) = f(x + 2) - 1 that corresponds to the known input of 33 for ff.
Set the argument of ff in g(x)g(x) equal to 33: x+2=3x + 2 = 3, which simplifies to x=1x = 1.
This allows us to substitute the known value f(3)f(3) into the expression for g(x)g(x).
3
Evaluate g(1)g(1) using the value of f(3)f(3).
g(1)=f(1+2)−1=f(3)−1=−4−1=−5g(1) = f(1 + 2) - 1 = f(3) - 1 = -4 - 1 = -5.
Substituting f(3)=−4f(3) = -4 into the simplified expression gives the output value of gg at x=1x = 1.
4
State the resulting point on the graph of y=g(x)y = g(x).
The point is (1,−5)(1, -5).
An input of x=1x = 1 yields an output of y=−5y = -5 for the function gg.

Key Concept

Function transformations and their effects on individual coordinate points.
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Topics

Function Notation and Transformations

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