Question

1. both of these functions grow as x gets larger and larger. which function eventually exceeds the other?

$f(x)=4^x - 5$
$g(x)=5x$

1. is this function linear, quadratic, or exponential?

x	y
4	16
5	25
6	36
7	49

options:
linear
quadratic
exponential

Explanation:

First Question:

Step1: Analyze function types

$f(x)=4^x - 5$ is exponential; $g(x)=5x$ is linear.

Step2: Compare growth rates

Exponential functions grow faster than linear functions as $x\to\infty$. For large $x$, $4^x$ dominates, so $4^x -5$ will exceed $5x$.

Step1: Check relationship pattern

Observe the table: when $x=3$, $y=9=3^2$; $x=4$, $y=16=4^2$; $x=5$, $y=25=5^2$; $x=6$, $y=36=6^2$; $x=7$, $y=49=7^2$. This follows $y=x^2$, a quadratic form.

Step2: Confirm function type

Quadratic functions have the form $y=ax^2+bx+c$ (here $a=1, b=0, c=0$), matching the table values.

Answer:

$f(x)=4^x - 5$ eventually exceeds $g(x)=5x$ as $x$ gets larger.

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Second Question: