Question

15 which is a true comparison for the following functions?
$f(x) = 4x + 7$

$x$	$g(x)$
$1$	$1$
$3$	$-5$
$5$	$-11$

a) the rate of change of $f(x) >$ the rate of change of $g(x)$
b) $f(3) = g(3)$
c) the $y$-intercept of $f(x) <$ the $y$-intercept of $g(x)$
d) $f(-1) > g(-1)$

Explanation:

Step1: Find the rate of change of \(f(x)\)

The function \(f(x)=4x + 7\) is in the form \(y=mx + b\), where \(m\) is the rate of change. So, the rate of change of \(f(x)\) is \(m_f=4\).

Step2: Find the rate of change of \(g(x)\)

Use the formula for the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Take two points from the table of \(g(x)\), say \((x_1=-1,y_1 = 7)\) and \((x_2=1,y_2 = 1)\). Then \(m_g=\frac{1 - 7}{1-(-1)}=\frac{-6}{2}=-3\).

Step3: Check option A

Since \(4>-3\) (i.e., \(m_f>m_g\)), option A is correct. But let's check other options for thoroughness.

Step4: Check option B

Find \(f(3)\): \(f(3)=4\times3 + 7=12 + 7=19\). From the table, \(g(3)=-5\). So \(f(3)
eq g(3)\).

Step5: Check option C

For \(f(x)=4x + 7\), the \(y\) - intercept \(b_f = 7\) (when \(x = 0\)). For \(g(x)\), use the point - slope form \(y - y_1=m(x - x_1)\). Using \(m_g=-3\) and \((x_1=-1,y_1 = 7)\), \(y-7=-3(x + 1)\), \(y-7=-3x-3\), \(y=-3x + 4\). The \(y\) - intercept \(b_g = 4\). So \(b_f>b_g\).

Step6: Check option D

Find \(f(-1)\): \(f(-1)=4\times(-1)+7=-4 + 7 = 3\). From the table \(g(-1)=7\). So \(f(-1)<g(-1)\).

Answer:

A. the rate of change of \(f(x)> \)the rate of change of \(g(x)\)