Question 446·Easy·Math

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The function $g$ is defined by

g(x)=3x^{\tfrac{3}{2}}

For what value of $x$ does $g(x)=24$ ?

A known output and an unknown input mean you're working backward through a function: set its rule equal to the output. A rational exponent such as $3/2$ combines a square root and a cube, so isolating the power doesn't yet isolate $x$ . A Desmos regression can find the input; don't mistake the value of the power for the input itself.

Hints

Hint 1
In function notation, $g(x)$ is the output produced by input $x$ . You know the output is $24$ , so set the rule equal to $24$ . What equation does that give you?
Hint 2
The denominator of the rational exponent $3/2$ means square root, and the numerator means cube. After dividing by $3$ , you know the value of $x^{3/2}$ , not the value of $x$ .

Step-by-step

Set the output and solve in Desmos
Step 1
Turn the output into an equation
The function notation $g(x)$ means the output produced by input $x$ . The given output is $24$ , so replace $g(x)$ with its rule: $3x^{3/2}=24$ .
Step 2
Isolate the power of $x$
Divide both sides by $3$ to isolate the power of $x$ : $x^{3/2}=8$ . The $8$ is what $x^{3/2}$ equals, not what $x$ equals.
Step 3
Read the fractional exponent
A rational exponent is a fraction used as an exponent. Its denominator gives the root, and its numerator gives the power. Here, $x^{3/2}=(\sqrt{x})^3$ . The square root requires $x\ge 0$ , so there's no negative real input to consider.
Step 4
Find the input in Desmos
Type $g(x)=3x^{3/2}$ , then $g(x_1)\sim24$ . Desmos uses $x$ to graph, so $x_1$ stands for the unknown input; $\sim$ tells it to find a value that gives the specified output. Under PARAMETERS, Desmos shows $x_1=4$ . So $g(x)=24$ when $x=4$ . Choice B.

Question 446·Easy·Math

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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