The graph of function f is given below. On the same grid plot: a) -f(2x) - 3; b) 2f(x - 4); c) 1/2 f(-1/2 x) - 6; d) f^-1.

Steps to Solve

1. Understanding the Transforms Each transformation represents a specific change to the function $f$. We need to analyze each part (a through d) individually to see how they affect the graph.

2. Transform for part (a): $-f(2x) - 3$

   • The $f(2x)$ indicates a horizontal compression by a factor of 2.
   • The $-f(2x)$ reflects the graph over the x-axis.
   • The $-3$ translates it downward by 3 units.

3. Transform for part (b): $2f(x - 4)$

   • The $f(x - 4)$ translates the graph to the right by 4 units.
   • The $2f(x-4)$ stretches the graph vertically by a factor of 2.

4. Transform for part (c): $\frac{1}{2}f\left(-\frac{1}{2}x\right) - 6$

   • The $-\frac{1}{2}x$ reflects the graph over the y-axis and compresses it horizontally by a factor of 2.
   • The $\frac{1}{2}f$ part vertically compresses the graph by a factor of 2.
   • The $-6$ translates it downward by 6 units.

5. Transform for part (d): $f^{-1}$

   • The $f^{-1}$ represents the inverse of the function, implying a reflection over the line $y = x$.

6. Graphing All Transforms

   • For each transformed function, plot it on the same grid as the original function $f$ to observe the differences and transformations clearly.