Question

Please help
maths functions

Answer 1

Explanation:

don't panic ! use your common sense and use the function to get the answers.

what don't you understand about functions ?

a function is a "translation" from an input value into an output value.

and "f(x)" is the same as saying "y", it just tells us to use "x" as input variable (if it is unspecified x), or to use a specified value in the function for "x".

this is like codes in riddles or secret messages, where you have to take a letter of the actual message, have to look it up in the coding table, and see what letter it should actually be. or as encoder, you have an actual letter, and you need to find out via the coding tables, what letter you should generate in the coded message.

a)

this here works exactly the same.

OA is the negative solution of what x was, when y was calculated to 0.

OB is the positive solution for the same situation.

it is, by the way, a property of a quadratic function (including a x² term as highest exponent), that any possible y value has normally 2 different x values as "source", and therefore also y = 0.

let's see ;

0 = 4 - x²

x² = 4

x = ±2

remember, a square root has 2 solutions, a positive and a negative one, as the square of a negative number is positive.

so,

OA = |-2| = 2 (as a distance is always positive)

OB = |+2| = 2

now to OH :

this means we need to calculate y, when x = 0.

now that is really simple :

y = 4 - 0² = 4

OH = 4

b)

OF = 1, and we need to find EF, which is the functional value f(x) for the position F, which means x = 1, as OF = 1.

remember, O = (0, 0).

so,

f(1) = y = 4 - 1² = 4 - 1 = 3

EF = 3

c)

this is the trickiest part.

but it follows the same principles.

as we said above, every possible y value has normally 2 x values as "source".

in a) we dealt with y = 0, and here we have now y = -5.

because the distance of CD and GK are both 5, but they are directed down, below the x-axis, so, the y values are meant to be negative. hence -5.

so,

-5 = 4 - x²

-9 = -x²

9 = x²

x = ±3

as D is on the left side of the y-axis, it must have the negative x coordinate (-3), and K on the right side the positive x coordinate (3).

but as explained in a), a distance is always positive,

OC = OG = 3

Answer 2

a) OA = OB = 2 units, OH = 4 units

b) EF = 3 units

c) OC = OG = 3 units

Explanation:

Given function:

`f(x)=4-x^2`

Part (a)

Points A and B are the x-intercepts of the parabola.

To find the x-intercepts (when y = 0), set the function to zero and solve for x:

`\begin{aligned}f(x) & = 0\\\implies 4-x^2 & = 0\\x^2 & = 4\\√(x^2) & = √(4)\\x & = \pm 2\end{aligned}`

Therefore, A (-2, 0) and B (2, 0) so OA = 2 units and OB = 2 units.

Point H is the y-intercept of the parabola.

To find the y-intercept, substitute x = 0 into the function:

`\implies f(0)=4-(0)^2=4`

Therefore, H (0, 4) so OH = 4 units.

Part (b)

If OF = 1 unit then point F is (-1, 0).

To find the y-value of point E, substitute x = -1 into the function:

`\implies f(-1)=4-(-1)^2=3`

Therefore, E is (-1, 3) so EF = 3 units.

Part (c)

If CD = GK = 5 units then the y-value of points C and G is -5.

To find the x-value of points C and G, set the function to -5 and solve for x:

`\begin{aligned}f(x) & = -5\\\implies 4-x^2 & = -5\\x^2 & = 9\\√(x^2) & = √(9)\\x & = \pm 3\end{aligned}`

Therefore, C (-3, 0) and G (3, 0) so OC = 3 units and OG = 3 units.