Question

question 36 · 1 point
given the following functions, compute the composition $(f \circ g)(x)$.
$f(x) = 8x + 9$
$g(x) = 7x - 8$
simplify your answer and collect like terms.
provide your answer below:
$f(g(x)) = \square$
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question 37 · 1 point
after 2 minutes, a submarine had descended to $-310$ feet. after 6 minutes, the submarine had descended to $-350$ feet. assuming a linear function, write an equation in the form $d(t) = mt + b$ that shows the depth, $d(t)$, after $t$ minutes.
provide your answer below:
$d(t) = \square$

Explanation:

Question 36

Step1: Recall function composition

Function composition \((f \circ g)(x) = f(g(x))\), so substitute \(g(x)\) into \(f(x)\).
\(f(g(x)) = f(7x - 8)\)

Step2: Substitute into \(f(x)\)

Since \(f(x) = 8x + 9\), replace \(x\) with \(7x - 8\):
\(f(7x - 8) = 8(7x - 8) + 9\)

Step3: Distribute and simplify

Distribute the 8: \(8 \times 7x = 56x\), \(8 \times (-8) = -64\)
So, \(56x - 64 + 9\)
Combine like terms: \(-64 + 9 = -55\)
Thus, \(f(g(x)) = 56x - 55\)

Step1: Find the slope \(m\)

The slope formula is \(m = \frac{d_2 - d_1}{t_2 - t_1}\). Given \((t_1, d_1) = (2, -310)\) and \((t_2, d_2) = (6, -350)\)
\(m = \frac{-350 - (-310)}{6 - 2} = \frac{-350 + 310}{4} = \frac{-40}{4} = -10\)

Step2: Find the y-intercept \(b\)

Use the point-slope form \(d - d_1 = m(t - t_1)\) or substitute \(m\), \(t\), and \(d\) into \(d(t) = mt + b\). Let's use \(t = 2\), \(d = -310\), \(m = -10\)
\(-310 = -10(2) + b\)
\(-310 = -20 + b\)
Add 20 to both sides: \(b = -310 + 20 = -290\)

Step3: Write the equation

Substitute \(m = -10\) and \(b = -290\) into \(d(t) = mt + b\):
\(d(t) = -10t - 290\)

Answer:

\(56x - 55\)

Question 37