Respuesta :
Using linear and composite functions, it is found that the x-intercept of g(h(x)) is [tex]x = -\frac{5}{6}[/tex], given by option B.
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The equation of a line is:
[tex]y = mx + b[/tex]
In which:
- m is the slope.
- b is the y-intercept.
Function g:
- Two of the points are (-6,16) and (-4,10).
- The slope is given by change in y divided by change in x, thus:
[tex]m = \frac{10 - 16}{-4 - (-6)} = -3[/tex]
Thus:
[tex]g(x) = -3x + b[/tex]
Point (-4,10) means that when [tex]x = -4, g(x) = 10[/tex], and we use this to find b, so:
[tex]10 = -3(-4) + b[/tex]
[tex]12 + b = 10[/tex]
[tex]b = -2[/tex]
Then
[tex]g(x) = -3x - 2[/tex]
Function h:
- Two of the points are (-6,-11) and (-4,-7).
- The slope is:
[tex]m = \frac{-7 - (-11)}{-4 - (-6)} = \frac{4}{2} = 2[/tex]
Thus:
[tex]h(x) = 2x + b[/tex]
Point (-4,-7) means that when [tex]x = -4, h(x) = -7[/tex], and we use this to find b, so:
[tex]-7 = 2(-4) + b[/tex]
[tex]-8 + b = -7[/tex]
[tex]b = 1[/tex]
Then
[tex]h(x) = 2x + 1[/tex]
The composite function is:
[tex]g(h(x)) = g(2x + 1) = -3(2x + 1) - 2 = -6x - 3 - 2 = -6x - 5[/tex]
The x-intercept it the zero of the function, thus:
[tex]-6x - 5 = 0[/tex]
[tex]6x = -5[/tex]
[tex]x = -\frac{5}{6}[/tex]
Which is option B.
A similar problem is given at https://brainly.com/question/21010520
