You can use grouping to isolate the variable in the equation \(mx+nx=p\)
\(mx+nx=p\) | Let's factor out an x from both sides terms on the left hand side of the equation. |
\(x(m+n)=p\) | Divide by m+n on both sides. |
\(x=\frac{p}{m+n}\) | |
The original expression is \(3^{-8}*3^5\). In order to do this, we can use the fact that the base of both exponents are the same and then simplify.
\(3^{-8}*3^5\) | Since the bases are identical, just add the exponents together. In general, \(a^b*a^c=a^{b+c}\) |
\(3^{-8}*3^5=3^{-8+5}=3^{-3}\) | A number to a negative exponent is the same as the inverse of a number to the same positive exponent. \(a^{-b}=\frac{1}{a^b}\) |
\(3^{-3}=\frac{1}{3^3}=\frac{1}{3*3*3}=\frac{1}{27}\) | This is your answer. |
The range is the set of all possible outputs that a function can produce.
\(f(x)=4|x+5|\)
The absolute value of any number will can result in a positive number or 0, and 4 multiplied by a positive number or 0 does not change this property at all. Therefore, the range is the following:
\(\text{Range}:\{\mathbb{R}|\hspace{1mm}y\geq0\}\)
This means that the range can be all nonnegative numbers. In interval notation, it would look like the following:
\(\text{Range}:[0,+\infty)\)
How do we find the intercepts? To do it without a graph, you can figure out by setting x=0 and y=0 and solving in each case. Let's do that.
\(y=4|x+5|\) | Plug in 0 for x. This time, we are solving for the y-intercept. |
\(y=4|0+5|\) | Simplify inside the absolute value first. |
\(y=4|5|\) | |
\(y=4*5=20\) | We have now determined the coordinates of the y-intercept. |
\((0,20)\) | This is the exact coordinates of the y-intercept. |
Let's do the exact same process. This time, however, we set y=0 to find the x-intecept.
\(y=4|x+5|\) | Set y equal to 0. |
\(0=4|x+5|\) | Divide by 4 on both sides. |
\(0=|x+5|\) | The absolute value always splits an equation into its plus or minus. However, 0 is neither positive nor negative, so there aren't 2 equations that one can set up. |
\(x+5=0\) | |
\(x=-5\) | We have now determined the x-intercept, as well. |
\((-5,0)\) | |
A constant term is the term that is unchanging. 6, for example, is a constant. 12.5 is a constant; they don't change.
To figure out \(f(g(x))\), let's break this down bit by bit. Since we already know that \(g(x)=x^3+5x^2+9x-2\), this means that \(f(g(x))=f(x^3+5x^2+9x-2)\).
\(f(x)=x^3-6x^2+3x-4\) | Now, substitute f(g(x)) into all instances of x. |
\(f(g(x))=(x^3+5x^2+9x-2)^3-6(x^3+5x^2+9x-2)^2+3(x^3+5x^2+9x-2)-4\) | Luckily, however, we only care about the constant terms. Let's deal with one term at a time. |
\((x^3+5x^2+9x-2)^3\) | We only care about the constant term, so do -2^3=-8 |
\((-2)^3=-8\) | Let's worry about the second term. |
\(-6(x^3+5x^2+9x-2)^2\) | Let's do the exact same process. |
\(-6*(-2)^2=-6*4=-24\) | And of course, the next term, as well. |
\(3(x^3+5x^2+9x-2)\) | |
\(3*-2=-6\) | And the final term, which happens to be a constant. |
\(-4=-4\) | Now, add all of those together to get the constant term. |
\(-8-24-6-4 = -42\) | This is value of the constant term. |