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Stage 4 - Stage 5

Evaluating Exponential Functions



Exponents and Indices

The idea of using superscripts to represent powers of numbers is not so old. In 1487 the French physician Nicolas Chuquet began the practise in his work Triparty, where he wrote expressions like $12^2$122 to represent what we would write as $12x^2$12x2

The idea of superscripted numbers, now called indices (singular index) or exponents, caught on. Writing expressions like $a^2$a2 meaning the square of $a$a, and $x^3$x3 meaning the cube of $a$a, were common place in the mathematical literature. 

Fairly soon, rules of indices were developed such as the five shown here:

Today, the mathematical world has become a lot more abstract, and the indices $m$m and $n$n are no longer restricted to integers. The change has occurred because of the increased interest in functions involving exponentials.

We therefore need to be able to evaluate expressions like $\sqrt{2}^{\sqrt{3}}$23 and $2^{-\sqrt{5}}$25 etc. and calculators have become indispensable tools for such tasks. 

Electronic calculators (and computer software) in the main show decimal approximations of expressions involving surds. For example, the calculator shows $1.822634654966242$1.822634654966242, an expression correct to $15$15 decimal places for the irrational number $\sqrt{2}^{\sqrt{3}}$23. Of course an infinite amount of digits is needed to express it exactly.

Finding exponential function values 

Exponential functions model an enormous number of physical phenomena including population growth, radioactive decay, compound interest and annuities, the shape of hanging ropes and chains and many other applications.

We are almost always asked to evaluate and simplify function values. Here are some examples to consider:


Example 1

If $f\left(x\right)=2^{-x}$f(x)=2x then $f\left(-3\right)=2^{-\left(-3\right)}=2^3=8$f(3)=2(3)=23=8.

Example 2

If $f\left(x\right)=\left(\frac{3}{2}\right)^{x+1}$f(x)=(32)x+1, then $f\left(-2\right)=\left(\frac{3}{2}\right)^{-2+1}=\left(\frac{3}{2}\right)^{-1}=\frac{2}{3}$f(2)=(32)2+1=(32)1=23 

Example 3

If $f\left(x\right)=5^x+5^{-x}$f(x)=5x+5x, then:

$f\left(-x\right)$f(x) $=$= $5^{-x}+5^{-\left(-x\right)}$5x+5(x)
  $=$= $5^{-x}+5^x$5x+5x
  $=$= $f\left(x\right)$f(x)

This shows that the curve of the function is symmetrical about the $y$y axis. Note that its lowest point is given by $f\left(0\right)=5^0+5^{-0}=2$f(0)=50+50=2.

Example 4

If $f\left(x\right)=3-2^x$f(x)=32x and $g\left(x\right)=3^{-x}$g(x)=3x, show that the difference between the composite function values $f\left(g\left(2\right)\right)$f(g(2)) and $g\left(f\left(2\right)\right)$g(f(2)) is given by $\sqrt[9]{2}$92

To answer this we derive an expression for the composite function  $f\left(g\left(x\right)\right)$f(g(x)):

$f\left(g\left(x\right)\right)$f(g(x)) $=$= $f\left(3^{-x}\right)$f(3x)
  $=$= $3-2^{\left(3^{-x}\right)}$32(3x)

We then form $g\left(f\left(x\right)\right)$g(f(x)) as follows:

$g\left(f\left(x\right)\right)$g(f(x)) $=$= $g\left(3-2^x\right)$g(32x)
  $=$= $3^{-\left(3-2^x\right)}$3(32x)
  $=$= $3^{\left(2^x-3\right)}$3(2x3)

Thus, $f\left(g\left(2\right)\right)=3-2^{\left(3^{-2}\right)}=3-\sqrt[9]{2}$f(g(2))=32(32)=392 and $g\left(f\left(2\right)\right)=3^{2^2-3}=3^1=3$g(f(2))=3223=31=3.

The difference between the two composite functions at $x=2$x=2 is clearly $\sqrt[9]{2}$92, which by a calculator is approximately $1.080059738892306$1.080059738892306

More Worked Examples


Find the value of $\left(\sqrt{3}\right)^{\sqrt{10}}$(3)10 correct to three decimal places.


Consider the function $f\left(x\right)=3^x+3^{-x}$f(x)=3x+3x.

  1. Evaluate $f\left(3\right)$f(3).

  2. Evaluate $f\left(-3\right)$f(3).

  3. Is $f\left(3\right)=f\left(-3\right)$f(3)=f(3)?






If $f\left(x\right)=5^x$f(x)=5x and $g\left(x\right)=3^{-x}$g(x)=3x, evaluate:

  1. $f\left(1\right)$f(1)

  2. $g\left(f\left(1\right)\right)$g(f(1))

  3. $f\left(g\left(0\right)\right)$f(g(0))

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