Question

1. both of these functions grow as x gets larger and larger. which function eventually exceeds the other? $f(x) = 2x + 10$ $g(x) = 3^{x} - 6$ submit
2. is this function linear, quadratic, or exponential?

x

y

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0

-2

1

-4

2

-8

3

-16

4

-32

linearquadraticexponential

Explanation:

First Question (Function Growth Comparison)

Step1: Test small x values

Calculate $f(x)$ and $g(x)$ for small $x$:

• $x=0$: $f(0)=2(0)+10=10$, $g(0)=3^0-6=1-6=-5$
• $x=3$: $f(3)=2(3)+10=16$, $g(3)=3^3-6=27-6=21$
• $x=4$: $f(4)=2(4)+10=18$, $g(4)=3^4-6=81-6=75$

Step2: Analyze long-term growth

Linear function $f(x)$ has constant slope $2$; exponential function $g(x)=3^x-6$ has growth rate that accelerates as $x$ increases. Exponential functions always outpace linear functions as $x\to\infty$.

Step1: Check linearity (constant slope)

Slope between $(0,-2)$ and $(1,-4)$: $\frac{-4-(-2)}{1-0}=-2$
Slope between $(1,-4)$ and $(2,-8)$: $\frac{-8-(-4)}{2-1}=-4$
Slopes are not constant, so not linear.

Step2: Check exponential (constant ratio)

Ratio of $y$-values for $x$ increasing by 1:
$\frac{-4}{-2}=2$, $\frac{-8}{-4}=2$, $\frac{-16}{-8}=2$, $\frac{-32}{-16}=2$
There is a constant multiplicative ratio of 2.

Answer:

The function $g(x)=3^x - 6$ eventually exceeds $f(x)=2x + 10$.

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Second Question (Function Type Identification)