Functions - Inverse Functions

Category: Algebra

"Published in Suisun City, California, USA"

Find the inverse function for

Solution:

Consider the given equation above

We know that y = f(x). Substitute y = f(x) as follows

Solve for x for the equation above

Replace x with y and y with x to the above equation, we have

Therefore,

Proving - Congruent Triangles, 2

Category: Plane Geometry

"Published in Newark, California, USA"

In the given figure, if MR ≅ PN; NR ≅ MP, prove that ∆NOR
≅ ∆MOP.

Photo by Math Principles in Everyday Life

Solution:

 Consider the given figure above

Photo by Math Principles in Everyday Life

Proof:

1. Statement: MR ≅ PN and NR ≅ MP.

    Reason: Given items.

2. Statement: PR ≅ PR

    Reason: Reflexive property of congruence.

3. Statement: ∆MRP ≅ ∆NPR

    Reason: Side Side Side (SSS) Postulate.

4. Statement: ∠MPR ≅ ∠NRP
                      ∠PMR ≅ ∠PNR
                      ∠PRM ≅ ∠RPN

    Reason: Since ∆MRP ≅ ∆NPR, then all interior angles of a triangle are congruent to all interior angles of other triangle. 

5. Statement: ∠PRM ≅ ∠4 and ∠RPN ≅ ∠3.

    Reason: Reflexive property of congruence.

6. Statement: ∠PRM ≅ ∠RPN ≅ ∠3 ≅ ∠4

    Reason: Transitive property of congruence.

7. Statement: ∆POR is an isosceles triangle.

    Reason: The two angles of an isosceles triangle are congruent. Hence, ∠3 ≅ ∠4 at the base.

8. Statement: OP ≅ OR

    Reason: Since ∆POR is an isosceles triangle, then the two sides of an isosceles triangle are congruent.

9. Statement: ∠MPR = ∠1 + ∠3 and ∠NRP = ∠2 + ∠4

    Reason: Addition property of angles.

10. Statement: ∠1 ≅ ∠2

      Reason: If ∠MPR ≅ ∠NRP and ∠3 ≅ ∠4, then it follows that ∠1 ≅ ∠2.

11. Statement: ∆MOP ≅ ∆NOR 

      Reason: Side Angle Side (SAS) Postulate.

Word Problem - Number Problem, 2

Category: Algebra

"Published in Newark, California, USA"

Determine the value of a 3-digit number if the unit's digit is 5 more than the hundred's digit and the ten's digit is one more than twice the hundred's digit if the sum of the digits is 2 more than twice the unit's digit.

Solution:

The given word problem above is about getting the value of a three digit number with given conditions for each digit. Let's analyze the given word problem as follows:

Let x = be the value of hundred's digit
      x + 5 = be the value of unit's digit
      2x + 1 = be the value of ten's digit

If the given word problem says ".....if the sum of the digits is 2 more than twice the unit's digit." then the working equation will be

Solve for the value of x. This will be the value of hundred's digit.

The value of hundred's digit = x = 3

The value of unit's digit = x + 5 = 3 + 5 = 8

The value of ten's digit = 2x + 1 = 2(3) + 1 = 6 + 1 = 7

Therefore, the value of a three digit number is 378.