If `f(x)` is a linear function such that `f(0)=-3 and f(2)=7,` find the formula for `f(x)`.

Answer

To find the formula for the linear function \( f(x) \), we start with the general form of a linear function:

\[ f(x) = mx + c \]

where \( m \) is the slope and \( c \) is the y-intercept.

### Step 1: Identify the y-intercept
From the problem, we know that \( f(0) = -3 \). This means that when \( x = 0 \), \( f(x) \) equals the y-intercept \( c \).

\[ f(0) = c = -3 \]

So, we can substitute \( c \) into our function:

\[ f(x) = mx - 3 \]

### Step 2: Use the second point to find the slope
Next, we use the second point given, \( f(2) = 7 \). We can substitute \( x = 2 \) into our function to find \( m \):

\[ f(2) = m(2) - 3 = 7 \]

### Step 3: Solve for \( m \)
Now, we can solve for \( m \):

\[ 2m - 3 = 7 \]

Add 3 to both sides:

\[ 2m = 10 \]

Now, divide by 2:

\[ m = 5 \]

### Step 4: Write the final function
Now that we have both \( m \) and \( c \), we can write the final formula for \( f(x) \):

\[ f(x) = 5x - 3 \]

### Final Answer
Thus, the formula for the linear function \( f(x) \) is:

\[ f(x) = 5x - 3 \]

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