Question

Find \(f \circ g \circ h\).

44. \(f\left( x \right) = {\bf{tan}}x\), \(g\left( x \right) = \frac{x}{{x - {\bf{1}}}}\), \(h\left( x \right) = \sqrt({\bf{3}}){x}\)

Short answer

The function is \(\tan \left( {\frac{{\sqrt(3){x}}}{{\sqrt(3){x} - 1}}} \right)\).

Step-by-step solution

Find the function \(g \circ h\)

The function \(g \circ h\left( x \right)\) can be calculated as follows:

\(\begin{aligned}g \circ h\left( x \right) &= g\left( {h\left( x \right)} \right)\\ &= g\left( {\sqrt(3){x}} \right)\\ &= \frac{{\sqrt(3){x}}}{{\sqrt(3){x} - 1}}\end{aligned}\)

Thus, the function is \(g \circ h\left( x \right) = \frac{{\sqrt(3){x}}}{{\sqrt(3){x} - 1}}\).

Find the function \(f \circ g \circ h\)

The function \(f \circ g \circ h\left( x \right)\) can be calculated as follows:

\(\begin{aligned}f^\circ g^\circ h\left( x \right) &= f\left( {\frac{{\sqrt(3){x}}}{{\sqrt(3){x} - 1}}} \right)\\ &= \tan \left( {\frac{{\sqrt(3){x}}}{{\sqrt(3){x} - 1}}} \right)\end{aligned}\)

So, the composite function is \(\tan \left( {\frac{{\sqrt(3){x}}}{{\sqrt(3){x} - 1}}} \right)\).