A known input in function notation calls for evaluation: replace the rule’s variable with the number in parentheses. Define the function in Desmos, then evaluate it by name. To check a fractional exponent, take the root indicated by its bottom number and the power indicated by its top number. A negative exponent means a reciprocal, not a negative value. It affects only the powered base, not a coefficient outside it.
Hints
- Hint 1
The input is the number inside a function’s parentheses. In , replace with . Which part of the resulting expression is raised to the negative power?
- Hint 2
In a fractional exponent of , the means square root and the means cube. What do you get by cubing ?
- Hint 3
A negative exponent turns the powered base into a reciprocal. It does not make the value negative or flip the separate coefficient . Once you know , what is ?
Step-by-step
Evaluate the function and check the exponent
Step 1Put the input into the rule
In , is the input, so it replaces in the given rule: . Type in Desmos to define the function for evaluation. The stays outside the power.
- Step 2
Evaluate the positive fractional power
The fractional exponent says to take a square root, then cube it. Rewrite the power: . Since , cube the result:
- Step 3
Apply the negative exponent
A negative exponent means a reciprocal. Flip the positive power: . Use the value you found: . The minus sign does not make the result negative.
- Step 4
Read the function’s output
Type below the definition. Desmos shows ; tap the fraction button to read . This checks : the multiplies the reciprocal rather than being flipped with it. So . Choice C.