The problem asks to determine which of the given equations defines $g(x)$ in terms of $f(x)$ based on the graphs.

Let's analyze the graphs:

1. Transformation between $f(x)$ and $g(x)$:

   • The graph of $g(x)$ appears to be a vertical stretch and a horizontal stretch (or compression) of the graph of $f(x)$.
   • Additionally, there might be a vertical shift involved.

2. Possible Transformations:

   • Vertical Scaling: The graph of $g(x)$ is stretched vertically compared to $f(x)$. This suggests that $g(x)$ could be some multiple of $f(x)$.
   • Horizontal Scaling: The graph of $g(x)$ seems horizontally compressed compared to $f(x)$, implying that $f(x)$ might be evaluated at a multiple of $x$.
   • Vertical Shift: There could be a shift downwards, as $g(x)$ seems to be shifted slightly compared to $f(x)$.

Now, let's analyze the given options:

• Option A: $g(x) = \frac{5}{2} f(2x)$

  • This suggests a vertical stretch by $\frac{5}{2}$ and a horizontal compression by $2$ (as multiplying $x$ by $2$ compresses the graph horizontally).

• Option B: $g(x) = \frac{5}{2} f\left(\frac{1}{2}x\right)$

  • This implies a vertical stretch by $\frac{5}{2}$ and a horizontal stretch by a factor of $2$ (as multiplying $x$ by $\frac{1}{2}$ stretches the graph horizontally).

• Option C: $g(x) = 3f(2x) - 1$

  • This involves a vertical stretch by $3$, a horizontal compression by $2$, and a vertical shift downwards by $1$.

• Option D: $g(x) = 3 f\left(\frac{1}{2}x\right) - 1$

  • This involves a vertical stretch by $3$, a horizontal stretch by $2$, and a vertical shift downwards by $1$.

Conclusion:

The correct answer is C: $g(x) = 3f(2x) - 1$, because:

• The graph of $g(x)$ is vertically stretched by a factor of 3 (compared to $f(x)$),
• Horizontally compressed by a factor of 2,
• And shifted down by 1 unit.

Would you like more details on any specific transformation or have any questions?

Related Questions:

1. How do horizontal stretches and compressions affect the graph of a function?
2. What is the effect of vertical translations on a graph?
3. Why does multiplying $f(x)$ by a constant stretch or compress it vertically?
4. How can we verify a horizontal compression from the graph?
5. What transformations combine to give the effect seen in the graph of $g(x)$?

Tip:

Always consider both horizontal and vertical changes when analyzing transformations between two functions' graphs!