randFromArray([["Even", $._("Even")], ["Odd", $._("Odd")], ["Neither", $._("Neither")]])
makeFunc(SOL_KEY) getUseablePoints(FUNC, SOL_KEY) randFromArray(X_VALUES)

According to the visible range of the graph, is `f(x)` even, odd, or neither?

`f(x)` is graphed below.

graphInit({ range: 10, scale: 20, tickStep: 1, axisArrows: "<->" }); style({ stroke: BLUE }); plot(function(x) { return FUNC(x); }, [-10, 10]); style({ stroke: RED });

A function is even if `f(x) = f(-x)` for all values of `x`.
A function is odd if `f(x) = -f(-x)` for all values of `x`.
style( { strokeWidth: 2 }, function() { path([[ PT, 0], [ PT, FUNC(PT)]]); path([[-PT, 0], [-PT, FUNC(-PT)]]); } ); style( { strokeDasharray: "." }, function() { path([[0, FUNC(PT)], [PT, FUNC(PT)]]); path([[0, FUNC(-PT)], [-PT, FUNC(-PT)]]); } );

`\qquad f(\blue{PT}) \approx \red{roundTo(1, FUNC(PT))}`
`\qquad f(\blue{-PT}) \approx \red{roundTo(1, FUNC(-PT))}`

`f(\blue{PT}) \neq f(\blue{-PT})`, so `f(x)` is not even.

`f(\blue{PT}) \neq -f(\blue{-PT})`, so `f(x)` is not odd.

Therefore `f(x)` is neither.

style( { strokeWidth: 2 }, function() { path([[ x, 0], [ x, FUNC(x)]]); path([[-x, 0], [-x, FUNC(-x)]]); } ); style( { strokeDasharray: "." }, function() { path([[0, FUNC(x)], [x, FUNC(x)]]); path([[0, FUNC(-x)], [-x, FUNC(-x)]]); } );

`\qquad f(\blue{x}) \approx \red{roundTo(1, FUNC(x))}`
`\qquad f(\blue{-x}) \approx \red{roundTo(1, FUNC(-x))}`

`f(x)` is odd because `f(x) = -f(-x)`. (For all x values, not just the ones we checked!)

`f(x)` is even because `f(x) = f(-x)`. (For all x values, not just the ones we checked!)

makeEquation(SOL_KEY) getNegativeFunction(FUNC) function(x){ return FUNC.evaluate(x); } getUseablePoints(CALLABLE_FUNC, SOL_KEY) FUNC.terms.length !== 1 || FUNC.terms[0].variableString !== ''

Is `f(x)` even, odd, or neither?

`f(x) = FUNC`

A function is even if `f(-x) = f(x)` for all values of `x`.
A function is odd if `f(-x) = -f(x)` for all values of `x`.

What is `f(\blue{-x})`?

`f(\blue{-x}) = NEG_FUNC1`

`f(\blue{-x}) = NEG_FUNC2`

`f(\blue{-x}) = -(FUNC)`

`f(\blue{-x}) = -f(x)`

Therefore, `f(x)` is odd.

`f(\blue{-x}) = f(x)`

Therefore, `f(x)` is even.

`f(-x) \neq f(x)`, since the signs of the terms with odd powers are different. `f(-x) \neq -f(x)` since the signs of the terms with even powers are the same.

Therefore, `f(x)` is neither odd nor even.

SOL_TEXT

- Even
- Odd
- Neither