Question

Maths functions question

Answer 1

a) OA = 1 unit

b) OB = 3 units

c) AB = √10 units

Explanation:

Given function:

`g(x)=2^x`

Part (a)

Point A is the y-intercept of the exponential curve (so when x = 0).

To find the y-value of Point A, substitute x = 0 into the function:

`\implies g(0)=2^0=1`

Therefore, A (0, 1) so OA = 1 unit.

Part (b)

If BC = 8 units then the y-value of Point C is 8.

The find the x-value of Point C, set the function to 8 and solve for x:

`\begin{aligned}f(x) & = 8 \\\implies 2^x & = 8\\2^x & = 2^3\\\implies x &= 3\end{aligned}`

Therefore, C (3, 8) so Point B is (3, 0). Therefore, OB = 3 units.

Part (c)

From parts (a) and (b):

• A = (0, 1)
• B = (3, 0)

To find the length of AB, use the distance between two points formula:

`d=√((x_2-x_1)^2+(y_2-y_1)^2)`

`\textsf{where }(x_1,y_1) \textsf{ and }(x_2,y_2)\:\textsf{are the two points.}`

Therefore:

`\implies \sf AB=√((x_B-x_A)^2+(y_B-y_A)^2)`

`\implies \sf AB=√((3-0)^2+(0-1)^2)`

`\implies \sf AB=√((3)^2+(-1)^2)`

`\implies \sf AB=√(9+1)`

`\implies \sf AB=√(10)\:\:units`