Question

complete the table of values for ( f(x) = 6x + 3 ) and ( g(x) = 5x^2 - x ). both ( f(x) ) and ( g(x) ) grow as ( x ) gets larger and larger. which function eventually exceeds the other? ( f(x) = 6x + 3 ) ( g(x) = 5x^2 - x )

Explanation:

Step1: Calculate f(x) for x=1

Substitute x=1 into $f(x)=6x+3$:
$6(1)+3=9$

Step2: Calculate f(x) for x=2

Substitute x=2 into $f(x)=6x+3$:
$6(2)+3=15$

Step3: Calculate f(x) for x=3

Substitute x=3 into $f(x)=6x+3$:
$6(3)+3=21$

Step4: Calculate f(x) for x=4

Substitute x=4 into $f(x)=6x+3$:
$6(4)+3=27$

Step5: Calculate g(x) for x=1

Substitute x=1 into $g(x)=5x^2-x$:
$5(1)^2-1=4$

Step6: Calculate g(x) for x=2

Substitute x=2 into $g(x)=5x^2-x$:
$5(2)^2-2=18$

Step7: Calculate g(x) for x=3

Substitute x=3 into $g(x)=5x^2-x$:
$5(3)^2-3=42$

Step8: Compare long-term growth

Quadratic functions grow faster than linear functions.

Answer:

Completed Table:

$x$	$f(x)$	$g(x)$
2	15	18
3	21	42
4	27	76

Growth Conclusion:

$g(x) = 5x^2 - x$ eventually exceeds $f(x) = 6x + 3$ as $x$ increases.