For the numbers $2$ through $9$ on a telephone keypad (see figure), create two relations: one mapping numbers onto letters, and the other mapping letters onto numbers. Are both relations functions? Explain.

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A relation that assigns to each element $x$ from a set of inputs, or $\underline{\hspace{1.5cm}}$ , exactly one element $y$ in a set of outputs, or $\underline{\hspace{1.5cm}}$, is called a $\underline{\hspace{1.5cm}}$.

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Horizontal shifts, vertical shifts, and reflections are called $\underline{\hspace{1.5cm}}$ transformations.

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Two techniques for fitting models to data are called direct $\underline{\hspace{1.5cm}}$ and least squares $\underline{\hspace{1.5cm}}$.

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If the composite functions $f(g(x))$ and $g(f(x))$ both equal $x$, then the function $g$ is the $\underline{\hspace{1.5cm}}$ function of $f$.

Find exact values of the inverse trigonometric functions or relations.
Arcsec $2$

Recursion relations to obtain the following recursion relations for spherical Bessel functions. We have written them for $j_n$, but they are valid for $y_n$ and for the $h_n’s.$ $(d / d x) j_{n}(x)=j_{n-1}(x)-(n+1) j_{n}(x) / x$

Use the recursion relations to obtain the following recursion relations for spherical Bessel functions. We have written them for $j_n$, but they are valid for $y_n$ and for the $h_n’s$. $(d / d x)\left[x^{-n} j_{n}(x)\right]=-x^{-n} j_{n+1}(x)$

Use the recursion relations to obtain the following recursion relations for spherical Bessel functions. We have written them for $j_n$, but they are valid for yn and for the $h_n’s$. $(d / d x)\left[x^{n+1} j_{n}(x)\right]=x^{n+1} j_{n-1}(x)$

List the 16 different relations on the set {0, 1}.

Find exact values of the inverse trigonometric functions or relations.
Arcsin $2$

Verify the angular momentum commutation relations in Eqs. we saw earlier.

Find exact values of the inverse circular functions or relations.\

$$\arctan 1$$

Why are supplier relations particularly important in a lean production environment?