Question 770·Easy·Math

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The function $g$ is defined by $g(x)=2\sqrt[3]{x}-1$ . For what value of $x$ does $g(x)=5$ ?

Given a function output and asked for its input, use inverse operations to work backward through the rule. Set the rule equal to the output; a Desmos regression can find the unknown input directly. Undo outside operations before clearing a root. A cube root is cleared by cubing, not squaring.

Hints

Hint 1
In function notation, $g(x)$ names the output you get from an input $x$ . Here you know the output, so replace $g(x)$ with its full rule and set that rule equal to the given output.
Hint 2
A regression lets Desmos find the input that makes an equation true. Type the function first, then use $x_1$ instead of $x$ in the equation so Desmos solves for a number rather than graphing.

Step-by-step

Approach 1: Find the input with Desmos

Step 1
Set the rule equal to the output
$g(x)$ is the output produced by input $x$ . Since the given output is $5$ , replace $g(x)$ with its rule:
$2\sqrt[3]{x}-1=5.$
Step 2
Read the input Desmos finds
Type $g(x)=2\sqrt[3]{x}-1$ , then $g(x_1)\sim5$ . The regression symbol $\sim$ asks Desmos to find the unknown input $x_1$ ; Desmos would treat plain $x$ as a graph variable. Under PARAMETERS, it shows $x_1=27$ . So the input that makes $g(x)=5$ is $x=27$ . Choice C.

Approach 2: Undo the function by hand

Step 1
Undo the subtraction
The $-1$ is the last operation in the rule, so reverse it first. Add $1$ to both sides of $2\sqrt[3]{x}-1=5$ :
$2\sqrt[3]{x}=6.$
Step 2
Isolate the cube root
The $2$ is a coefficient, a number multiplying the cube root. Divide both sides by $2$ :
$\sqrt[3]{x}=3.$
Step 3
Cube to recover the input
Cube both sides because cubing undoes a cube root:
$x=3^3.$
Evaluate the cube:
$x=27.$
Squaring $3$ would give $9$ , but squaring doesn't undo a cube root. The input is $27$ . Choice C.

Question 770·Easy·Math

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Step-by-step

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Strategy

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