Which of the following represents the graph of a function?

Here are the solutions to the problems:

46. Which of the following represents the graph of a function?
A graph represents a function if it passes the vertical line test, meaning any vertical line intersects the graph at most once.

• Graph (A) fails the vertical line test.
• Graph (B) fails the vertical line test.
• Graph (C) passes the vertical line test.
• Graph (D) fails the vertical line test. The correct option is (C).

$$\boxed{\text{C)}}$$

47. The straight line AB cuts the Y axis at
From the graph provided for Item 47, the line intersects the Y-axis at the point where $x=0$. Observing the graph, this point is $(0, -1)$.
The correct option is (D).

$$\boxed{\text{D) } (0, -1)}$$

48. The gradient of AB in the graph above is
From the graph provided for Item 48, we can identify two points on the line AB.
Point A appears to be $(2, 1)$.
Point B appears to be $(6, 3)$.
The formula for the gradient $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Substitute the coordinates of A and B:

$$m = \frac{3 - 1}{6 - 2}$$ $$m = \frac{2}{4}$$ $$m = \frac{1}{2}$$

The correct option is (B).

$$\boxed{\text{B) } \frac{1}{2}}$$

49. The inequality above is defined by
The diagram shows a number line with a shaded region.
There is an open circle at $-2$, indicating that $-2$ is not included in the solution set.
There is a closed circle at 3, indicating that 3 is included in the solution set.
The shaded region is between $-2$ and 3.
Therefore, the inequality is $-2 < x \le 3$.
The correct option is (B).

$$\boxed{\text{B) } -2 < x \le 3}$$

50. If $P = 2x^2 + 2x - 1$, then $P(1) =$
To find $P(1)$, substitute $x=1$ into the expression for $P$.

$$P(1) = 2(1)^2 + 2(1) - 1$$ $$P(1) = 2(1) + 2 - 1$$ $$P(1) = 2 + 2 - 1$$ $$P(1) = 4 - 1$$ $$P(1) = 3$$

The correct option is (D).

$$\boxed{\text{D) } 3}$$

51. What is the gradient of the line $2y - 8x = 7$?
To find the gradient, rearrange the equation into the slope-intercept form $y = mx + c$, where $m$ is the gradient.

$$2y - 8x = 7$$

Add $8x$ to both sides of the equation:

$$2y = 8x + 7$$

Divide both sides by 2:

$$y = \frac{8x}{2} + \frac{7}{2}$$ $$y = 4x + \frac{7}{2}$$

The gradient $m$ is the coefficient of $x$.

$$m = 4$$

The correct option is (D).

$$\boxed{\text{D) } 4}$$