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Answer:
The order from least to greatest is: h(x), g(x), f(x)
This is true for x = 3, x = 6, and x = 15
The function values are given in a table as shown in the attached image.
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Explanation:
For each function, we replace x with the specified value. Then we compare the outputs to see which is larger.
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Let's plug x = 3 into each function
Start with f(x)
f(x) = x^2 + 10x - 5
f(3) = (3)^2 + 10(3) - 5
f(3) = 34 ... use a calculator, or PEMDAS
Repeat for g(x)
g(x) = 8x+1
g(3) = 8(3)+1
g(3) = 25
and then h(x)
h(x) = 3x-4
h(3) = 3(3)-4
h(3) = 5
The order from least to greatest is h(x), g(x), f(x)
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Let's plug x = 6 into each function
Start with f(x)
f(x) = x^2 + 10x - 5
f(6) = (6)^2 + 10(6) - 5
f(6) = 91
Repeat for g(x)
g(x) = 8x+1
g(6) = 8(6)+1
g(6) = 49
and then h(x)
h(x) = 3x-4
h(6) = 3(6)-4
h(6) = 14
The order from least to greatest is h(x), g(x), f(x)
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You don't need to plug x = 15 into each function to know that h is the smallest, followed by g, then f being the largest. This is because the linear functions always have the slowest growth compared to quadratic functions. So f(x) being quadratic is the largest output of all three. The smaller of the linear outputs happens with the smaller slope. Recall that slope is rate of change telling us how fast a function grows (assuming the slope is positive). We see that h(x) has slope 3 compared to g(x) that has a slope of 8
Check out the attached image below. Note how everything to the right of point C, with x coordinate 1.65, is where the functions do not cross one another anymore. Beyond this point, h(x) is always the smallest with slowest growth, followed by g(x) in the middle, and then f(x) is the fastest growing function with the largest outputs for any one single input.
