To determine whether the functions $f$ and $g$ are linear or exponential, let’s analyze the table of values provided.

Function $f(x)$:

The values of $f(x)$ are:
$6, 12, 18, 24$ for $x = 1, 2, 3, 4$.

• Difference between consecutive $f(x)$:
  $12 - 6 = 6$,
  $18 - 12 = 6$,
  $24 - 18 = 6$.
  The differences are constant.
• This means $f(x)$ is linear.

Consecutive differences are constant at 6.

Function $g(x)$:

The values of $g(x)$ are:
$6, 12, 24, 48$ for $x = 1, 2, 3, 4$.

• Ratios between consecutive $g(x)$:
  $\frac{12}{6} = 2$,
  $\frac{24}{12} = 2$,
  $\frac{48}{24} = 2$.
  The ratios are constant.
• This means $g(x)$ is exponential.

Consecutive ratios are constant at 2.

Final Answers:

1. Function $f$:

   • The function $f$ is linear.
   • Consecutive differences are 6.

2. Function $g$:

   • The function $g$ is exponential.
   • Consecutive ratios are 2.

Related Questions:

1. What is the general formula for a linear function?
2. How can you determine if a function is exponential using its table of values?
3. What are the properties of an exponential function's ratios?
4. If the function $f(x)$ were $f(x) = 3x + 3$, what would its table look like?
5. Can you write the equation for $g(x)$ given it is exponential with a base of 2?

Tip: For a function to be exponential, the ratio between consecutive outputs must remain constant. For a function to be linear, the differences between consecutive outputs must remain constant.