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Given the function $\text{\hspace{0.17em}}h\left(p\right)={p}^{2}+2p,\text{\hspace{0.17em}}$ evaluate $\text{\hspace{0.17em}}h\left(4\right).\text{\hspace{0.17em}}$
To evaluate $\text{\hspace{0.17em}}h\left(4\right),\text{\hspace{0.17em}}$ we substitute the value 4 for the input variable $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ in the given function.
Therefore, for an input of 4, we have an output of 24.
Given the function $\text{\hspace{0.17em}}g\left(m\right)=\sqrt{m-4},\text{\hspace{0.17em}}$ evaluate $\text{\hspace{0.17em}}g\left(5\right).$
$\text{\hspace{0.17em}}g\left(5\right)=1\text{\hspace{0.17em}}$
Given the function $\text{\hspace{0.17em}}h\left(p\right)={p}^{2}+2p,\text{\hspace{0.17em}}$ solve for $\text{\hspace{0.17em}}h\left(p\right)=3.$
If $\text{\hspace{0.17em}}\left(p+3\right)\left(p-1\right)=0,\text{\hspace{0.17em}}$ either $\text{\hspace{0.17em}}\left(p+3\right)=0\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(p-1\right)=0\text{\hspace{0.17em}}$ (or both of them equal 0). We will set each factor equal to 0 and solve for $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ in each case.
This gives us two solutions. The output $\text{\hspace{0.17em}}h\left(p\right)=3\text{\hspace{0.17em}}$ when the input is either $\text{\hspace{0.17em}}p=1\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}p=-3.\text{\hspace{0.17em}}$ We can also verify by graphing as in [link] . The graph verifies that $\text{\hspace{0.17em}}h\left(1\right)=h\left(-3\right)=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(4\right)=24.$
Given the function $\text{\hspace{0.17em}}g\left(m\right)=\sqrt{m-4},\text{\hspace{0.17em}}$ solve $\text{\hspace{0.17em}}g\left(m\right)=2.$
$\text{\hspace{0.17em}}m=8\text{\hspace{0.17em}}$
Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation $\text{\hspace{0.17em}}2n+6p=12\text{\hspace{0.17em}}$ expresses a functional relationship between $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}p.\text{\hspace{0.17em}}$ We can rewrite it to decide if $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is a function of $\text{\hspace{0.17em}}n.\text{\hspace{0.17em}}$
Given a function in equation form, write its algebraic formula.
Express the relationship $\text{\hspace{0.17em}}2n+6p=12\text{\hspace{0.17em}}$ as a function $\text{\hspace{0.17em}}p=f\left(n\right),\text{\hspace{0.17em}}$ if possible.
To express the relationship in this form, we need to be able to write the relationship where $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is a function of $\text{\hspace{0.17em}}n,\text{\hspace{0.17em}}$ which means writing it as $\text{\hspace{0.17em}}p=[\text{expression}\text{\hspace{0.17em}}\text{involving}\text{\hspace{0.17em}}n].$
Therefore, $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is written as
Does the equation $\text{\hspace{0.17em}}{x}^{2}+{y}^{2}=1\text{\hspace{0.17em}}$ represent a function with $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ as input and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as output? If so, express the relationship as a function $\text{\hspace{0.17em}}y=f\left(x\right).$
First we subtract $\text{\hspace{0.17em}}{x}^{2}\text{\hspace{0.17em}}$ from both sides.
We now try to solve for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ in this equation.
We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function $\text{\hspace{0.17em}}y=f\left(x\right).$ If we graph both functions on a graphing calculator, we will get the upper and lower semicircles.
If $\text{\hspace{0.17em}}x-8{y}^{3}=0,\text{\hspace{0.17em}}$ express $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x.$
$y=f\left(x\right)=\frac{\sqrt[3]{x}}{2}$
Are there relationships expressed by an equation that do represent a function but that still cannot be represented by an algebraic formula?
Yes, this can happen. For example, given the equation $\text{\hspace{0.17em}}x=y+{2}^{y},\text{\hspace{0.17em}}$ if we want to express $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ there is no simple algebraic formula involving only $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that equals $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ However, each $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ does determine a unique value for $\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}$ and there are mathematical procedures by which $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ even though the formula cannot be written explicitly.
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