Question

From the list below, indicate which exponential function represents an increasing function. Select all that apply.• h (x)=7(0.9)^x• k (x)=10(3/5)^x• n (x)=4(7/6)^x• p (x)= -10(8)^x• j (x)=2(1+0.03)^x• g (x)=0.25×6^x• f (x)=5×2^x• m (x)=3(4)^x -5

Answer 1

Exponential functions

Initial explanation

An exponential function is given by the formula:

`f(x)=a\cdot b^x`

where a and b are numbers. b is always positive

We have that there are two ways of obtaining a decreasing exponential function:

1. if a is negative

2. if 0 < b < 1

We have that we have an increasing function if and only if

a is positive and b is higher than 1, b > 1

Analysis

We have that:

In h(x) = 7 · 0.9ˣ

a = 7 and b = 0.9

Since 0 < 0.9 < 1, then it is a decreasing function.

In k(x) = 10 · (3/5)ˣ

a = 10 and b = 3/5 = 0.6

Since 0 < 3/5 < 1, then it is a decreasing function.

In n(x) = 4 · (7/6)ˣ

a = 4 and b = 7/6 = 1.166...

Since a is positive and b is higher than 1: 1 < 1.166...,

then it is an increasing function.

In p(x) = -10 · 8ˣ

a = -10 and b = 8

Since a is negative, then it is a decreasing function.

In j(x) = 2 · (1 + 0.03)ˣ

a = 2 and b = 1 + 0.03 = 1.03

Since a is positive and b is higher than 1: 1 < 1.03,

then it is an increasing function.

In g(x) = 0.25 · 6ˣ

a = 0.25 and b = 6

Since a is positive and b is higher than 1: 1 < 6,

then it is an increasing function.

In f(x) = 5 · 2ˣ

a = 5 and b = 2

Since a is positive and b is higher than 1: 1 < 2,

then it is an increasing function.

In m(x) = 3 · 4ˣ - 5

a = 3 and b = 4

Since a is positive and b is higher than 1: 1 < 4,

then it is an increasing function.

Answer- the increasing functions are: