Is this the correct graph for the piecewise function f(x)=⎧⎩⎨⎪⎪2x+6 if x<1−14x−3 if 1≤x<64+4x if x≥6

Question

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Answer 1

To determine if a graph correctly represents the given piecewise function, we first need to analyze the pieces of the function defined for different ranges of \( x \).

The function \( f(x) \) is defined as follows: \[ f(x) = \begin{cases} 2x + 6 & \text{if } x < 1 \ -\frac{1}{4}x - 3 & \text{if } 1 \leq x < 6 \ 4 + 4x & \text{if } x \geq 6 \end{cases} \]

Let’s break down each interval:

1. For \( x < 1 \): The function is \( f(x) = 2x + 6 \).

   • This is a linear function with a slope of 2 and a y-intercept of 6.

2. For \( 1 \leq x < 6 \): The function is \( f(x) = -\frac{1}{4}x - 3 \).

   • This is also a linear function, but it has a slope of -1/4 and a y-intercept of -3.

3. For \( x \geq 6 \): The function is \( f(x) = 4 + 4x \).

   • This is a linear function with a slope of 4 and a y-intercept of 4.

Now, at the boundaries of the intervals, we can also check the values of the function:

• At \( x = 1 \):

  • For \( x < 1 \), \( f(1) = 2(1) + 6 = 8 \).
  • For \( 1 \leq x < 6 \), \( f(1) = -\frac{1}{4}(1) - 3 = -\frac{1}{4} - 3 = -\frac{13}{4} or -3.25\).

  The function has a jump discontinuity at \( x = 1 \), since the left-hand limit does not equal the function value at \( x = 1 \).

• At \( x = 6 \):

  • For \( 1 \leq x < 6 \), \( f(6) = -\frac{1}{4}(6) - 3 = -\frac{3}{2} - 3 = -\frac{9}{2} \ or -4.5\).
  • For \( x \geq 6 \), \( f(6) = 4 + 4(6) = 4 + 24 = 28\).

  There's another jump discontinuity at \( x = 6 \).

If the graph you have does not show these characteristics (i.e., linear segments, jumps at the boundaries, etc.), then it is likely incorrect. To verify whether your graph correctly represents the function, check these features against it:

• The slope and intercepts for each piece should match the calculations.
• There should be open or closed circles at the endpoints of the intervals according to whether that endpoint is included in the interval (closed circle for included, open circle for excluded).
• Significant jumps at \( x = 1 \) and \( x = 6\).

If your graph matches all of these criteria, then it is indeed the correct representation of the piecewise function.