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Given the graph of the function f(x)=|x+4|

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(1 point)
Responses

{(3,−7),(2,−6),(1,−5),(−4,0),(1,−3),(2,−2),(3,−1)}
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{(3,−7),(2,−6),(1,−5),(0,0),(1,−3),(2,−2),(3,−1)}
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1 answer

To find the ordered pairs that would be used to graph the inverse of the function \( f(x) = |x + 4| \), we should first identify the points on the graph of the original function and then switch the x-coordinates and y-coordinates to get the points for the inverse function.

  1. Identify points on \( f(x) = |x + 4| \):

    • The vertex occurs at \( x = -4 \). When \( x = -4 \), \( f(-4) = 0 \), giving us the point \( (-4, 0) \).
    • For \( x = -3 \), \( f(-3) = 1 \) giving the point \( (-3, 1) \).
    • For \( x = -2 \), \( f(-2) = 2 \) giving the point \( (-2, 2) \).
    • For \( x = -1 \), \( f(-1) = 3 \) giving the point \( (-1, 3) \).
    • For \( x = 0 \), \( f(0) = 4 \) giving the point \( (0, 4) \).
    • The function is symmetric about its vertex, so likewise for \( x = 1 \), \( f(1) = 5 \), giving point \( (1, 5) \), and so forth.

  2. Finding the inverse: The inverse function \( f^{-1}(y) \) is found by switching \( x \) and \( y \) in the original points:

    • From \( (-4, 0) \) we get \( (0, -4) \).
    • From \( (-3, 1) \) we get \( (1, -3) \).
    • From \( (-2, 2) \) we get \( (2, -2) \).
    • From \( (-1, 3) \) we get \( (3, -1) \).
    • From \( (0, 4) \) we get \( (4, 0) \), and so on.

Now, let's verify the given options:

The options provided were:

  1. \( {(3,−7),(2,−6),(1,−5),(−4,0),(1,−3),(2,−2),(3,−1)} \)
  2. \( {(3,−7),(2,−6),(1,−5),(0,0),(1,−3),(2,−2),(3,−1)} \)

None of these are complete matches for the set of points from the original function, but we identify \( (1,-3) \), \( (2,-2) \), and \( (3,-1) \) in both sets, along with \( (-4,0) \) in the first set, which corresponds correctly to the vertex of the absolute value function.

To summarize, the points \( (1, -3) \), \( (2, -2) \), and \( (3, -1) \) correspond to the inverse function, while \( (-4, 0) \) is the vertex and should also be present but isn't exactly part of the ordered inversion.

Given this analysis, our best options related to the inverse pairs based on the vertex point would lead us towards selecting the first set.

Therefore, the ordered pairs used to graph the inverse would be: \[ {(3,−7),(2,−6),(1,−5),(−4,0),(1,−3),(2,−2),(3,−1)} \]