Chapter 2.4, Problem 23E

To determine

To state:the domain and explain by using theorem 4,5,7,9 why the function iscontinuous at every number in its domain.

Answer to Problem 23E

  $\left(1,\infty \right)$ .

Explanation of Solution

Given:

  $G\left(t\right)=\ln \left(t^4-1\right)$ .

Concept used:

Theorem 4 - the quotient of continuous functions is also continuous.

Theorem 7 - polynomials and logarithm functions are continuous.

Theorem 9-if $f\left(x\right)$ and $g\left(x\right)$ are continuous, the composite function is also continuous

Calculation:

The given function is $G\left(t\right)=\ln \left(t^4-1\right)$ .

Now, according to theorem-7, $\left(t^4-1\right)\text{ and}\ln \left(t\right)$ are continuous.

Also, $\ln \left(t^4-1\right)$ is continuous according to theorem 9.

Therefore, the domain is $\left(1,\infty \right)$ since, the function $t\le 1,\text{ then}{t}^4-1\le 0$ the logarithm function is undefined for non-positive values.