Question

Exercise 2.5: Function and exponents 1. Graph the functions a) y = 16 + 2x b) y = 8 – 2x c) y = 2x +12 (In each case, consider the domain as consisting of nonnegative real numbers only) 5. Condense the following expressions: b) x^a\times x^b\times x^c c){\ \ x}^3\times y^3\times z^3 6. Find: b) \left(x^{1/2}\times x^{1/3}\right)/x^{2/3}

Answer 1

Solution:

1) Graphing the functions:

The best way to graph a linear function is simply finding the horizontal (x-axis) and vertical (y-axis) intercepts, and joining the two. Since we are considering domain as consisting of non-negative real number only, we are concerned only with the first quadrant of XY-plane.

a) y = 16 + 2x

y - 2x = 16

Dividing whole expression by 16, (-2x/16) + (y/16) = 1

Or: x/(-8) + y/(16) = 1

So, we have horizontal intercept as (-8) and vertical intercept as 16. Following is the required graph:

40 y- 16+2x 36 32 28 24 20 12 8 642 246 810 12 14 1618 20

b) y = 8 - 2x

2x + y = 8

Dividing whole expression by 8: x/4 + y/8 = 1

So, we have horizontal intercept as 4 and vertical intercept as 8. Following is the required graph:

c) y = 2x + 12

-2x + y = 12

Dividing whole expression by 12: x/(-6) + y/12 = 1

So, we have horizontal intercept as (-6) and vertical intercept as 12. Following is the required graph:

2) Condensing the expressions: A term written as X^Y denotes X as the base and Y as the exponent or power. Basic rules for condensing of multiplication and division, especially required for this question involves:

1) For the expression with terms with same base carrying exponents, the expression can be shortened as base raised to the power of addition of the exponents of terms which are multiplied and subtraction of the exponents of terms which are divided.

2) For the expression with terms with same exponents over say different bases, the expression can be shortened as taking the power common for all, and writing the bases together as originally multiplied or divided.

This will soon become more clear:

b) $x^a\times x^b \times x^c$ = $x^{a+b+c}$ (using rule (1) above)

c) $x^3\times y^3 \times z^3$ = $(x\times y \times z)^3$ (using rule (2) above)

6. Solving expressions:

We have to find: $\frac{x^{1/2}\times x^{1/3}}{x^{2/3}}$

Clearly, the expression has a common base with different powers. So, using rule number 1 from above:

$\frac{x^{1/2}\times x^{1/3}}{x^{2/3}}$ = x^{1/2 + 1/3 - 2/3} = x^(3+2-4)/6 = x^1/6