The table of values shown is for some linear function f(x). Find f(10).
`{:(x,y),(-2,-11),(-1,-7),(0,-3),(1,1):}`

Answer

To find \( f(10) \) for the linear function represented by the given table of values, we can follow these steps:

### Step 1: Identify the linear function form
The standard form of a linear function is:
\[ f(x) = mx + c \]
where \( m \) is the slope and \( c \) is the y-intercept.

### Step 2: Determine the y-intercept \( c \)
From the table, we can use the point where \( x = 0 \) and \( y = -3 \):
\[ f(0) = -3 \]
This means:
\[ c = -3 \]

### Step 3: Calculate the slope \( m \)
To find the slope \( m \), we can use any two points from the table. Let's use the points \( (-1, -7) \) and \( (1, 1) \).

The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the points:
\[ m = \frac{1 - (-7)}{1 - (-1)} = \frac{1 + 7}{1 + 1} = \frac{8}{2} = 4 \]

### Step 4: Write the linear function
Now that we have both \( m \) and \( c \), we can write the function:
\[ f(x) = 4x - 3 \]

### Step 5: Calculate \( f(10) \)
Now we can find \( f(10) \):
\[ f(10) = 4(10) - 3 = 40 - 3 = 37 \]

### Final Answer
Thus, \( f(10) = 37 \).

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