Question

x | f(x) = 5(x + 2)
0 | 10
1 | 15
2 | 20
3 | 25
4 | 30

x | m(x) = 3 - (1/2)x
0 | 3
1 | 2½
2 | 2
3 | 1½
4 | 1

Explanation:

Since the problem involves functions and their tabular representations (like \( f(x) = 5(x + 2) \) and \( m(x)=3-\frac{1}{2}x \)) with input - output pairs, it falls under the sub - field of Algebra in Mathematics. Let's assume we want to find when \( f(x)=m(x) \) (a common algebraic problem with functions).

Step 1: Write the equations

We have \( f(x)=5(x + 2)=5x + 10 \) and \( m(x)=3-\frac{1}{2}x \). To find the value of \( x \) where \( f(x)=m(x) \), we set the two equations equal to each other:
\( 5x+10 = 3-\frac{1}{2}x \)

Step 2: Add \(\frac{1}{2}x\) to both sides

Adding \(\frac{1}{2}x\) to both sides of the equation \( 5x + 10=3-\frac{1}{2}x \) gives us:
\( 5x+\frac{1}{2}x+10=3-\frac{1}{2}x+\frac{1}{2}x \)
Simplifying the left - hand side (LHS) and right - hand side (RHS), we know that \( -\frac{1}{2}x+\frac{1}{2}x = 0 \), and \( 5x+\frac{1}{2}x=\frac{10x + x}{2}=\frac{11x}{2} \). So the equation becomes \( \frac{11x}{2}+10 = 3 \)

Step 3: Subtract 10 from both sides

Subtract 10 from both sides of the equation \( \frac{11x}{2}+10 = 3 \):
\( \frac{11x}{2}+10 - 10=3 - 10 \)
Simplifying, we get \( \frac{11x}{2}=-7 \)

Step 4: Solve for x

Multiply both sides of the equation \( \frac{11x}{2}=-7 \) by \( \frac{2}{11} \):
\( x=-7\times\frac{2}{11}=-\frac{14}{11}\approx - 1.27 \)

If we check the tables, for \( x = 0 \), \( f(0)=10 \), \( m(0) = 3 \); \( x = 1 \), \( f(1)=15 \), \( m(1)=2\frac{1}{2}\); \( x = 2 \), \( f(2)=20 \), \( m(2)=2 \); \( x = 3 \), \( f(3)=25 \), \( m(3)=1\frac{1}{2} \); \( x = 4 \), \( f(4)=30 \), \( m(4)=1 \). Since \( f(x) \) is increasing and \( m(x) \) is decreasing, and at \( x = 0 \), \( f(x)>m(x) \), and our solution for \( f(x)=m(x) \) is at \( x=-\frac{14}{11}\), which is less than 0, so in the domain of \( x\geq0 \) (as per the table), \( f(x)>m(x) \) for all \( x = 0,1,2,3,4 \)

Answer:

If we assume the question is to compare the functions in the table, for \( x = 0,1,2,3,4 \), \( f(x)>m(x) \). If we were to solve \( f(x)=m(x) \), \( x =-\frac{14}{11}\)