Question

find the domainand range of f(x) =√(9-x)​

Answer 1

We're given a function,

$\longrightarrow f(x)=\sqrt{9-x}$

We've to find its domain and range.

Since the function $f(x)=\sqrt x$ is defined only for $x\geq0,$ we get,

$\longrightarrow 9-x\geq0$

$\longrightarrow x\leq9$

Or,

$\longrightarrow\underline{\underline{x\in(-\infty,\ 9]}}$

This is the domain of the function.

Let,

$\longrightarrow y=\sqrt{9-x}$

$\longrightarrow y^2=9-x$

$\longrightarrow x=9-y^2\quad\quad\dots(1)$

Consider the domain of our function.

$\longrightarrow x\in(-\infty,\ 9]$

From (1),

$\longrightarrow 9-y^2\in(-\infty,\ 9]$

Adding 9,

$\longrightarrow -y^2\in(-\infty,\ 0]$

Multiplying by -1, [note the interval limit change]

$\longrightarrow y^2\in[0,\ \infty)$

Taking the square root,

$\longrightarrow y\in[0,\ \infty)$

[$y\notin(-\infty,\ 0)$ because $\sqrt{y^2}=|y|.$]

Or,

$\longrightarrow\underline{\underline{f(x)\in[0,\ \infty)}}$

This is the range of the function.

Answer 2

We're given a function,

\longrightarrow f(x)=\sqrt{9-x}⟶f(x)=

9−x

We've to find its domain and range.

Since the function f(x)=\sqrt xf(x)=

x

is defined only for x\geq0,x≥0, we get,

\longrightarrow 9-x\geq0⟶9−x≥0

\longrightarrow x\leq9⟶x≤9

Or,

\longrightarrow\underline{\underline{x\in(-\infty,\ 9]}}⟶

x∈(−∞, 9]

This is the domain of the function.

Let,

\longrightarrow y=\sqrt{9-x}⟶y=

9−x

\longrightarrow y^2=9-x⟶y

2

=9−x

\longrightarrow x=9-y^2\quad\quad\dots(1)⟶x=9−y

2

…(1)

Consider the domain of our function.

\longrightarrow x\in(-\infty,\ 9]⟶x∈(−∞, 9]

From (1),

\longrightarrow 9-y^2\in(-\infty,\ 9]⟶9−y

2

∈(−∞, 9]

Adding 9,

\longrightarrow -y^2\in(-\infty,\ 0]⟶−y

2

∈(−∞, 0]

Multiplying by -1, [note the interval limit change]

\longrightarrow y^2\in[0,\ \infty)⟶y

2

∈[0, ∞)

Taking the square root,

\longrightarrow y\in[0,\ \infty)⟶y∈[0, ∞)

[y\notin(-\infty,\ 0)y∈

/

(−∞, 0) because \sqrt{y^2}=|y|.

y

2

=∣y∣.

Or,

\longrightarrow\underline{\underline{f(x)\in[0,\ \infty)}}⟶

f(x)∈[0, ∞)

This is the range of the function.