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

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?

1 answer

Let's first calculate the values of f(x)=3x+1 and g(x)=x2+5 for the given values of x.

Calculating Values

  1. For x=0:

    • f(0)=3(0)+1=1
    • g(0)=02+5=5
  2. For x=1:

    • f(1)=3(1)+1=4
    • g(1)=12+5=6
  3. For x=2:

    • f(2)=3(2)+1=7
    • g(2)=22+5=9
  4. For x=3:

    • f(3)=3(3)+1=10
    • g(3)=32+5=14

Filling the Table

Now we can complete the table:

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=x2+5

Rearranging gives us:

x23x+4=0

Calculating the discriminant:

D=b24ac=(3)24(1)(4)=916=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.