Function f is defined by `f(x)=2x+1`. If `2f(m)=30`, what is the value of `f(2m)`?

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Understand the function and the given equation
The function \( f \) is defined as:
\[ f(x) = 2x + 1 \]
We are also given that:
\[ 2f(m) = 30 \]

### Step 2: Simplify the equation
From the equation \( 2f(m) = 30 \), we can divide both sides by 2 to find \( f(m) \):
\[ f(m) = \frac{30}{2} = 15 \]

### Step 3: Substitute \( m \) into the function
Now we know that \( f(m) = 15 \). We can substitute \( m \) into the function:
\[ f(m) = 2m + 1 \]
Setting this equal to 15 gives us:
\[ 2m + 1 = 15 \]

### Step 4: Solve for \( m \)
To find \( m \), we will isolate it:
1. Subtract 1 from both sides:
\[ 2m = 15 - 1 \]
\[ 2m = 14 \]
2. Divide both sides by 2:
\[ m = \frac{14}{2} = 7 \]

### Step 5: Find \( f(2m) \)
Now we need to find \( f(2m) \). First, we calculate \( 2m \):
\[ 2m = 2 \times 7 = 14 \]
Now we substitute \( 14 \) into the function \( f \):
\[ f(2m) = f(14) = 2(14) + 1 \]
Calculating this gives:
\[ f(14) = 28 + 1 = 29 \]

### Final Answer
Thus, the value of \( f(2m) \) is:
\[ \boxed{29} \]