Question

1. consider the linear function $f(x)=7x+25$ and the exponential function $g(x)=4(1.35)^x$.

(a) how many units larger is the y-intercept of the linear function than that of the exponential function?
(b) explain how you know that each function is increasing by their equations.

Explanation:

Step1: Find f(x) y-intercept

Substitute $x=0$ into $f(x)$:
$f(0)=7(0)+25=25$

Step2: Find g(x) y-intercept

Substitute $x=0$ into $g(x)$:
$g(0)=4(1.35)^0=4(1)=4$

Step3: Calculate difference

Subtract the two intercepts:
$25-4=21$

Step4: Analyze linear function increase

Check slope of $f(x)$:
Slope $m=7>0$, so $f(x)$ increases.

Step5: Analyze exponential function increase

Check base of $g(x)$:
Base $1.35>1$, so $g(x)$ increases.

Answer:

(a) 21
(b) For the linear function $f(x)=7x+25$, its slope is 7, which is a positive value, so the function increases as $x$ increases. For the exponential function $g(x)=4(1.35)^x$, the base of the exponential term is 1.35, which is greater than 1, so the function increases as $x$ increases.