An exponential function in the form starts at when and multiplies by each time increases by . When a new function puts an expression inside , substitute that whole expression into the original rule. You can define both functions in Desmos and find the new starting value and multiplier; don’t assume the original base stays the same.
Hints
- Hint 1
In a function composition, everything inside is its input. Replace the exponent in the given rule with the whole . What power of does that produce?
- Hint 2
For an exponential function , the value at is because . Evaluate in Desmos to find the new coefficient.
- Hint 3
A multiplier tells you what happens to an output when increases by . The exponent then increases by , so what factor does the power of gain?
Step-by-step
Find the starting value and multiplier
Step 1Put the whole new input into the rule
Type , then in Desmos. The input to is its exponent, so replacing with the whole input gives . Both definitions are now ready for the evaluations below.
- Step 2
Find the new starting value
In , the coefficient is the value at , since . Type ; Desmos shows . So the new coefficient is , not : the in the exponent changes the starting value.
- Step 3
Find the multiplier for one step in x
When increases by , the exponent increases by . That multiplies the output by , rather than by . Type ; Desmos shows . Doubling inside the exponent squares the base’s one-step multiplier.
- Step 4
Write the rule for h
An exponential function with starting value and a constant one-step multiplier of has this rule: . It defines the requested function for every , not only the inputs you checked in Desmos. Choice C.
Lessons that teach this
- SAT Nonlinear FunctionsAdvancedCoreTransform nonlinear functions
- SAT Advanced AlgebraBeginnerCoreUse exponent rules and common bases
- SAT Nonlinear FunctionsIntermediateEvaluate nonlinear functions and recover inputs
- DesmosBeginnerCoreEvaluate, combine, and compose functions
- DesmosAdvancedCoreExponential models, transformations, and Log Mode