HiSET: Math : Evaluate functions for inputs in their domains

Study concepts, example questions & explanations for HiSET: Math

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Example Questions

Example Question #1 : Evaluate Functions For Inputs In Their Domains

Define \(\displaystyle f(x)= -\frac{1}{x+1}\) and \(\displaystyle g(x) = |x+2|\).

Evaluate \(\displaystyle (f\circ g )(2)\).

Possible Answers:

\(\displaystyle -\frac{1}{4}\)

\(\displaystyle -1\)

The expression is undefined.

\(\displaystyle \frac{1}{3}\)

\(\displaystyle - \frac{1}{5}\)

Correct answer:

\(\displaystyle - \frac{1}{5}\)

Explanation:

By definition,

\(\displaystyle (f\circ g )(2) = f(g(2))\),

so first, evaluate \(\displaystyle g(2)\) by substituting 2 for \(\displaystyle x\) in \(\displaystyle g(x)\):

\(\displaystyle g(x) = |x+2|\)

\(\displaystyle g(2) = |2+2| = |4| =4\)

\(\displaystyle (f\circ g )(2) = f(4)\),

so evaluate \(\displaystyle f(4)\) through a similar substitution:

\(\displaystyle f(x)= -\frac{1}{x+1}\)

\(\displaystyle f(4)= -\frac{1}{4+1} = - \frac{1}{5}\),

the correct response.

Example Question #1 : Evaluate Functions For Inputs In Their Domains

Define functions \(\displaystyle f\) and \(\displaystyle g\) as follows:

\(\displaystyle f(x)= \left\{\begin{matrix} x^{2} -2,&x< 0 \\ x+ 3,& x \ge 0\end{matrix}\right.\)

\(\displaystyle g(x)= x+ 4\)

Evaluate \(\displaystyle (f \circ g)(0)\).

Possible Answers:

Undefined

\(\displaystyle 14\)

\(\displaystyle 7\)

\(\displaystyle 2\)

\(\displaystyle 0\)

Correct answer:

\(\displaystyle 7\)

Explanation:

By definition, \(\displaystyle (f \circ g)(0) = f(g(0))\). Evaluate \(\displaystyle g(0)\) by substituting 0 for \(\displaystyle x\) in the definition of \(\displaystyle g\):

\(\displaystyle g(x)= x+ 4\)

\(\displaystyle g(0)= 0+ 4= 4\)

\(\displaystyle (f \circ g)(0) = f(g(0)) = f(4)\), so substitute 4 for \(\displaystyle x\) in the definition of \(\displaystyle f\). Since 4 satisfies the condition \(\displaystyle x \ge 0\), use the definition for those values:

\(\displaystyle f(x)= x+ 3\)

\(\displaystyle f(4)= 4+ 3 = 7\),

the correct value.

 

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