Question

if $f(x)=j\circ h$ then $f(1)=?$
answer
numeric 1 point
if $f(x)=h\circ g$ then $f(2)=?$
answer
numeric 1 point
if $f(x)=h\circ j$ then $f(3)=?$
for all questions use the following functions:
$g(x)=3x + 2$ $h(x)=\frac{2}{x}$ $j(x)=x^2 - 1$

Explanation:

First Sub - Question: If \(f(x)=j\circ h\) then \(f(1)=?\)

Step 1: Recall the definition of function composition

The composition of two functions \(j\) and \(h\) is defined as \((j\circ h)(x)=j(h(x))\). So we first need to find \(h(1)\) and then substitute it into \(j(x)\).

Step 2: Calculate \(h(1)\)

Given \(h(x)=\frac{2}{x}\), substitute \(x = 1\) into \(h(x)\): \(h(1)=\frac{2}{1}=2\).

Step 3: Calculate \(j(h(1))\)

Given \(j(x)=x^{2}-1\), substitute \(x = h(1)=2\) into \(j(x)\): \(j(2)=2^{2}-1=4 - 1=3\).

Step 1: Recall the definition of function composition

The composition of two functions \(h\) and \(g\) is defined as \((h\circ g)(x)=h(g(x))\). So we first need to find \(g(2)\) and then substitute it into \(h(x)\).

Step 2: Calculate \(g(2)\)

Given \(g(x)=3x + 2\), substitute \(x = 2\) into \(g(x)\): \(g(2)=3\times2+2=6 + 2=8\).

Step 3: Calculate \(h(g(2))\)

Given \(h(x)=\frac{2}{x}\), substitute \(x = g(2)=8\) into \(h(x)\): \(h(8)=\frac{2}{8}=\frac{1}{4}\).

Step 1: Recall the definition of function composition

The composition of two functions \(h\) and \(j\) is defined as \((h\circ j)(x)=h(j(x))\). So we first need to find \(j(3)\) and then substitute it into \(h(x)\).

Step 2: Calculate \(j(3)\)

Given \(j(x)=x^{2}-1\), substitute \(x = 3\) into \(j(x)\): \(j(3)=3^{2}-1=9 - 1=8\).

Step 3: Calculate \(h(j(3))\)

Given \(h(x)=\frac{2}{x}\), substitute \(x = j(3)=8\) into \(h(x)\): \(h(8)=\frac{2}{8}=\frac{1}{4}\).

Answer:

\(3\)

Second Sub - Question: If \(f(x)=h\circ g\) then \(f(2)=?\)