Math III
Go further with polynomial and rational expressions, advanced functions, trigonometry, geometric modeling, and statistical inference.
- Problem types
- 641
- Practice variants
- 2,564
Page 7 of 18
Each problem type has four distinct practice variants. Open a preview to move among all four.
Write a finite geometric series model for repeated depreciation or discount and identify the first term, common ratio, and number of terms
Derive and use the finite geometric series formula to solve problems such as mortgage-payment models.
Convert a percent loss into the fraction that remains, since repeated depreciation multiplies by the retention factor rather than by the loss rate. Timing determines the first exponent: if the …
Preview problemEvaluate and precisely interpret a finite geometric model
Derive and use the finite geometric series formula to solve problems such as mortgage-payment models.
Evaluation and interpretation are both part of using a financial model. Enter the full grouped expression, keep calculator precision through the exponent, subtraction, division, and multiplication, and round only the …
Preview problemDecide whether the finite geometric series formula applies for a given common ratio
Derive and use the finite geometric series formula to solve problems such as mortgage-payment models.
Always check the denominator condition built into a closed formula. When the common ratio makes that denominator zero, the usual geometric quotient is undefined even though the finite sum itself …
Preview problemCompare financial plans at a matched valuation date
Derive and use the finite geometric series formula to solve problems such as mortgage-payment models.
A fair financial comparison requires one shared valuation date. Accumulate each plan through its own last deposit, then carry any earlier-ending balance forward through the remaining periods even when no …
Preview problemForm a function sum and intersect the domains
Combine studied function types arithmetically to build models.
A function sum pairs the two outputs produced by the same input; it does not merge the input rules themselves. Write both expressions side by side, combine only like terms, …
Preview problemForm a function difference, domain, and sign comparison
Combine studied function types arithmetically to build models.
A difference of functions is an ordered comparison: evaluate both at the same input and subtract the entire second output from the first. Parentheses protect every sign when the second …
Preview problemForm a function product with domain and supported units
Combine studied function types arithmetically to build models.
Multiplying functions means multiplying their outputs at a shared input. Keep each expression grouped long enough to distribute every term across every term; that structure prevents a missing cross-product or …
Preview problemForm and interpret a contextual function quotient
Combine studied function types arithmetically to build models.
Average cost asks how much of the total cost belongs to one item, so divide the whole cost model by the production count. Splitting the quotient separates the fixed-cost share …
Preview problemBuild, unit, restrict, and analyze a combined contextual model
Combine studied function types arithmetically to build models.
The average-cost model separates into a constant per-item charge plus a fixed cost spread across all produced items. Increasing production makes only the fixed-cost share shrink, so the graph falls …
Preview problemForm and interpret a combined function model from two component functions
Combine studied function types arithmetically to build models.
Treat subtraction as a modeling instruction: at each time, compare the growth-model output with the resource-limit output in the stated order. The combined rule is meaningful only where both components …
Preview problemForm and evaluate an arithmetic combination of two functions
Combine studied function types arithmetically to build models.
A function name acts like an input-output machine, so feed the same requested input into each rule separately. Keep the two substitutions distinct until both outputs are known, then add …
Preview problemEvaluate \((f+g)(a)\) from given function values and interpret the combined output
Combine studied function types arithmetically to build models.
When individual function values are already given, those values are all the information needed for the combined output. Match both values to the same input, preserve the sign on each …
Preview problemDetermine the domain of a combined function by combining all input restrictions
Combine studied function types arithmetically to build models.
A combined expression is defined only where every component is defined, so find each restriction separately and intersect them. Square roots allow a zero radicand, while logarithms require a strictly …
Preview problemChoose the correct arithmetic operation to combine functions from a context
Combine studied function types arithmetically to build models.
A context determines the operation by the relationship between quantities, not merely by the fact that two functions appear. A total gathers contributions measured in the same units, whereas a …
Preview problemForm a function sum and verify its graph feature
Combine studied function types arithmetically to build models.
Build the combined model only after checking that the two components have compatible output units. Then keep the contextual domain beside the algebra: an item count is discrete and positive, …
Preview problemWrite and interpret a combined function model from a context
Combine studied function types arithmetically to build models.
Translate the story one charge at a time before combining anything: identify the constant contribution, the quantity that grows with speed, and the quantity divided by speed. Because each piece …
Preview problemRead signed horizontal and vertical shifts and verify an anchor
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
Read transformations from standard form, treating input changes and output changes differently. An expression inside the function moves horizontal coordinates with the opposite visible sign, while an outside addition moves …
Preview problemRead scales from the displayed A f(Bx) form
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
In the form A times f of B x, position determines which coordinate changes. The outside factor multiplies outputs, while the inside factor changes horizontal distances by its reciprocal. Write …
Preview problemSelect reflection axes and simplify the transformed function
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
First decide whether a negative changes the input or the entire output; its location determines which coordinate is negated. Write the corresponding point map, test it on a familiar parent …
Preview problemAnalyze a transformed radical graph with anchor, domain, and range
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
A square-root graph is organized around its endpoint, so move that anchor before describing the rest of the curve. The inside shift sets where allowed inputs begin, and the outside …
Preview problemAnalyze a transformed reciprocal parent and its asymptotes
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
A reciprocal graph is organized by its two parent asymptotes, so transform those guide lines before thinking about branch points. The denominator’s zero locates the excluded input and vertical boundary, …
Preview problemAnalyze a transformed exponential graph from a,b,h,k
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
An exponential transformation is easiest to track through its baseline and the parent anchor where the exponent is zero. Move that anchor according to the horizontal and vertical shifts, then …
Preview problemAnalyze a transformed logarithm and its domain boundary
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
A logarithm is organized around an excluded input boundary. Set its argument equal to zero to locate that boundary, then use the strict positivity requirement to determine which side belongs …
Preview problemWrite an equation for a transformed parent function from key graph features
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
Start with the transformed form for the identified parent family rather than guessing an equation from isolated coordinates. A square-root endpoint fixes both shift parameters immediately, leaving only the vertical …
Preview problemIdentify graph features after inside and outside additions in a transformed function
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
For a transformed square-root function, the endpoint provides both a geometric anchor and the boundary of the domain. Rewrite the inside expression in standard shift form to control its sign, …
Preview problemProve evenness using domain symmetry and f(-x)
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
An evenness proof has two gates: the domain must admit each input together with its opposite, and the outputs must satisfy the defining equality. Substitute negative x everywhere with parentheses …
Preview problemProve or disprove oddness using f(-x) and -f(x)
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
Testing oddness requires a symmetric domain and a precise comparison between two independently formed expressions. Compute f of negative x by replacing every input, then compute negative f of x …
Preview problemClassify a function as even, odd, both, or neither
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
Even and odd are separate definition tests, not automatically exclusive labels. After confirming domain symmetry, compare f of negative x first with f of x and then independently with negative …
Preview problemInterpret inside timing and outside baseline with units
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
Interpret a contextual transformation by keeping input roles and output roles separate. The expression inside the radical resets the time clock and locates when the model begins, while the outside …
Preview problemCompare transformed graphs in parallel feature rows
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
Compare transformed graphs in parallel rather than jumping directly from one equation to the other. Find each square-root endpoint and domain independently, then subtract corresponding endpoint coordinates to measure the …
Preview problemFind an inverse function by swapping x and y and solving a linear equation
Find inverse functions for simple invertible functions, including rational examples.
An inverse reverses the roles of input and output, so begin by writing the rule with y and swapping the variables. Then solve for the new output by undoing the …
Preview problemFind the inverse of a restricted quadratic by swapping x and y
Find inverse functions for simple invertible functions, including rational examples.
A quadratic becomes invertible only after its domain restriction keeps one branch. Record the original range before swapping variables, because that range becomes the inverse domain. When solving the swapped …
Preview problemFind an inverse function by swapping x and y and solving a square root equation
Find inverse functions for simple invertible functions, including rational examples.
Before inverting a square-root rule, record its domain and range so their exchange is not lost in the algebra. Swap input and output, isolate the radical, and square only after …
Preview problemFind the inverse of a cube-root function by swapping x and y and solving
Find inverse functions for simple invertible functions, including rational examples.
Cube-root functions are one-to-one across all real inputs, so their inverses need no branch decision. After swapping input and output, isolate the cube root and undo it by cubing the …
Preview problemFind an inverse function by swapping x and y and solving a rational equation
Find inverse functions for simple invertible functions, including rational examples.
For a shifted reciprocal, swap input and output first, then isolate the reciprocal expression before taking reciprocals. Reversing too early would apply the operation to a sum instead of the …
Preview problemFind an inverse function by swapping x and y and solving an exponential equation
Find inverse functions for simple invertible functions, including rational examples.
An exponential and a logarithm with the same base describe the same relationship from opposite directions. Swap the variables, then rewrite the exponential equation in logarithmic form so the unknown …
Preview problem