Complete the table of values for f(x)=3x+1 and g(x)=x2+5.

Question

x f(x) g(x)
0
1
2
3
Both f(x) and g(x) grow as x gets larger and larger. Which function eventually exceeds the other?

Answer 1

Let's first calculate the values of $f(x)=3x+1$ and $g(x)=x^2+5$ for the given values of $x$.

Calculating Values

1. For $x=0$:

   • $f(0)=3(0)+1=1$
   • $g(0)=0^2+5=5$

2. For $x=1$:

   • $f(1)=3(1)+1=4$
   • $g(1)=1^2+5=6$

3. For $x=2$:

   • $f(2)=3(2)+1=7$
   • $g(2)=2^2+5=9$

4. For $x=3$:

   • $f(3)=3(3)+1=10$
   • $g(3)=3^2+5=14$

Filling the Table

Now we can complete the table:

$$\text{Misplaced hline}$$

Analyzing Growth of Functions

Next, we need to determine which function eventually exceeds the other as $x$ becomes larger.

• $f(x)$ is a linear function, growing at a constant rate of 3.
• $g(x)$ is a quadratic function, which grows faster as $x$ increases because quadratic functions increase faster than linear functions for sufficiently large $x$.

To find out when $g(x)$ will exceed $f(x)$, we can set them equal to each other and solve for $x$:

$$3x+1=x^2+5$$

Rearranging gives us:

$$x^2-3x+4=0$$

Calculating the discriminant:

$$D=b^2-4ac=(-3{)}^2-4(1)(4)=9-16=-7$$

Since the discriminant is negative, this means the equation has no real solutions which indicates that $g(x)$ starts exceeding $f(x)$ at all $x$ greater than zero.

Conclusion

Thus, while both functions grow, $g(x)$ will eventually exceed $f(x)$ as $x$ increases, particularly noticeable after $x=3$.