Functions

[CAD60]

Can someone please help solve this problem

f(x) = 5x – 7

g(x) = 5x/(x + 4)

6(a) Write down the value of x that must be excluded from any domain of g.

6(b) Find gf(2.6)

6(c) Solve fg(x) = 2

6(d) Express the inverse function g^-1(x)

 

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[R.M.]

Part a: "... must be excluded from any domain..." is the same as saying "what values can you NOT plug into that function?"

 

[Otis]

Hi CAD60. I'm not familiar with the notation in parts (b) and (c).

I suppose that gf(2.6) could refer to either a composition of functions or a product of functions. Can you clarify?

For a composition, I've seen $\color{blue}(g \circ f)(2.6)\color{black}$ and
$\color{blue}g(f(2.6))\color{black}$.

See:

Composition of Functions

For a product, I've seen $\color{blue}(g \times f)(2.6)\color{black}$ and
$\color{blue}(g*f)(2.6)\color{black}$.

See:

Operations with Functions

 

[khansaheb]

Can someone please help solve this problem

Please plot g(x) for x = -5 to x=5. You may want to use "wolfram Alpha" to assist you in plotting.

Please share the plot with us.

 

[Steven G]

You can't compute square roots of negative numbers and you can't divide by 0.

g(x) has no square roots----so you have no chance of computing the square root of a negative number. The denominator, x+4, equals 0 when x=? Remove that number from the domain.

 

[Otis]

f(x) = 5x – 7
g(x) = 5x/(x + 4)

6(a) Write down the value of x that must be excluded from any domain of g

CAD60 has not responded. Here's my work for other readers.

Function g is a ratio of polynomials. All polynomials have the same domain: The set of Real numbers.

As Steven posted, the denominator in a ratio cannot be zero. Hence, the answer is -4.

6(b) Find gf(2.6)

If the notation denotes function composition, then the output of function f (at x=2.6) is used as the input to function g.

f(2.6) = 5(2.6) – 7 = 6

g(6) = 5(6)/(6 + 4) = 3

Therefore: g(f(2.6)) = 3

---

If the notation denotes a product of functions, then the outputs of functions f and g (at x=2.6) are multiplied together.

g(2.6) = 5(2.6)/(2.6 + 4) = 1.969696…

f(2.6) $\times$ g(2.6) = (6)(1.969696…) = 11.818181…

6(c) Solve fg(x) = 2

If the notation denotes function composition, then the output of function g is used as the input to function f.

f(g(x)) = $5\big(\color{brown}\frac{5x}{x+4}\color{black}\big)-7 = \frac{25x}{x+4}-7$

Hence, the equation to solve is:

$\displaystyle \frac{25x}{x+4}-7\,=\,2$

Add 7 and multiply by (x+4):

25x = 9(x + 4)
25x = 9x + 36
16x = 36
x = 9/4

---

If the notation denotes a product of functions, then f(x) is multiplied by g(x).

$\displaystyle (5x-7)\bigg(\frac{5x}{x+4}\bigg) = \frac{25x^{2}-35x}{x+4}$

Hence, the equation to solve is:

$\displaystyle \frac{25x^{2}-35x}{x+4}=2$

Multiply by (x+4), distribute and move all terms to the left-hand side side:

$\displaystyle 25x^{2}-37x-8 = 0$

The quadratic formula yields two solutions:

$\displaystyle x\,=\,\frac{37}{50}\pm\frac{3\sqrt{241}}{50}$

6(d) Express the inverse function g^-1(x)

To find function g's inverse, start with y=5x/(x+4) and swap symbols x and y:

x = 5y/(y + 4)

Next, solve for y:

Multiply each side by (y+4) and distribute
Subtract 5y and 4x from each side
Factor out y on the left-hand side
Divide each side by (x–5)

g^-1(x) = -4x/(x – 5)

The graphs of a function and its inverse are symmetric about the line y=x, so plotting all three with equal scales on the x- and y-axis is a good check. Here is such a graph of g(x), g^-1(x) and x near the origin.

 

[khansaheb]

CAD60 has not responded. Here's my work for other readers.

Function g is a ratio of polynomials. All polynomials have the same domain: The set of Real numbers.

As Steven posted, the denominator in a ratio cannot be zero. Hence, the answer is -4.


If the notation denotes function composition, then the output of function f (at x=2.6) is used as the input to function g.

f(2.6) = 5(2.6) – 7 = 6

g(6) = 5(6)/(6 + 4) = 3

Therefore: g(f(2.6)) = 3

---

If the notation denotes a product of functions, then the outputs of functions f and g (at x=2.6) are multiplied together.

g(2.6) = 5(2.6)/(2.6 + 4) = 1.969696…

f(2.6) $\times$ g(2.6) = (6)(1.969696…) = 11.818181…


If the notation denotes function composition, then the output of function g is used as the input to function f.

f(g(x)) = $5\big(\color{brown}\frac{5x}{x+4}\color{black}\big)-7 = \frac{25x}{x+4}-7$

Hence, the equation to solve is:

$\displaystyle \frac{25x}{x+4}-7\,=\,2$

Add 7 and multiply by (x+4):

25x = 9(x + 4)
25x = 9x + 36
16x = 36
x = 9/4

---

If the notation denotes a product of functions, then f(x) is multiplied by g(x).

$\displaystyle (5x-7)\bigg(\frac{5x}{x+4}\bigg) = \frac{25x^{2}-35x}{x+4}$

Hence, the equation to solve is:

$\displaystyle \frac{25x^{2}-35x}{x+4}=2$

Multiply by (x+4), distribute and move all terms to the left-hand side side:

$\displaystyle 25x^{2}-37x-8 = 0$

The quadratic formula yields two solutions:

$\displaystyle x\,=\,\frac{37}{50}\pm\frac{3\sqrt{241}}{50}$


To find function g's inverse, start with y=5x/(x+4) and swap symbols x and y:

x = 5y/(y + 4)

Next, solve for y:

Multiply each side by (y+4) and distribute
Subtract 5y and 4x from each side
Factor out y on the left-hand side
Divide each side by (x–5)

g^-1(x) = -4x/(x – 5)

The graphs of a function and its inverse are symmetric about the line y=x, so plotting all three with equal scales on the x- and y-axis is a good check. Here is such a graph of g(x), g^-1(x) and x near the origin.

View attachment 38249

In my opinion another good check would be to show that

g(g^-1(x)) = x ...... and
g^-1(g(x)) = x

 

[Otis]

In my opinion another good check would be to show that
g(g^-1(x)) = x ...

I agree! Please share your work, as an example for future readers.