Question

Let f(x) = Inx, g(x) = \xl and h(x) = sinx. The
range of (fogoh)(x) is​

Answer 1

Given:

Three functions:

$f(x) = lnx$

$g(x) = |x|$ and

$h(x) = sinx$

To find:

Range of the function:

$fogoh(x)$

Solution:

First of all, let us find the value of the composite function $fogoh(x)$, then we can talk about the range of it.

Let us proceed step by step.

First of all, let us find the value of $goh(x)$

$goh(x) = g(h(x))$

i.e. Replacing $x$ in $g(x)$ with $h(x) = sinx$, we get:

$goh(x) = g(h(x))=|sinx|$

Now, let us find the value of $fogoh(x)$ which will be nothing but:

$fogoh(x) = f(g(h(x)))$

i.e. replacing $x$ with $g(h(x))$ i.e. $|sinx|$, we get:

$fogoh(x) = f(g(h(x)))= ln(|sinx|)$

This composite function returns the natural log of $|sinx|$.

We know that $sinx$ always lies in the interval [-1, 1] and

So, $|sinx|$ will lie in the interval [0, 1]

And value of natural log lies between -$\infty$ to 0 in this interval.

So, the range of function $fogoh(x)$ is (-$\infty$, 0].